#72318754

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×#99
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁵
prime radical	#7632941	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#3428 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

5860 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,11)(4,5)(6,10)(7,9)(8,13)(12,16)(14,15)(17,18,23,27)(19,28,32,38)(20,25,22,37)(21,34,35,26)(24,30,40,29)(31,45,46,42)(33,47,36,41)(39,43,48,44);(1,3,6,13,7,12,5,14)(2,8,10,16,9,15,4,11)(17,19)(18,24)(20,30,22,29)(21,31)(23,32)(25,28,37,38)(26,39)(27,40)(33,44,36,43)(34,48)(35,46)(41,42,47,45);(2,9)(4,10)(8,15)(11,16)(17,20)(18,25)(19,29)(21,33)(22,23)(24,38)(26,41)(27,37)(28,40)(30,32)(31,43)(34,47)(35,36)(39,45)(42,48)(44,46);(1,4,7,10)(2,5,9,6)(3,8,12,15)(11,13,16,14)(19,30)(20,22)(21,35)(24,38)(25,37)(26,34)(28,40)(29,32)(31,43)(39,42)(44,46)(45,48);(2,9)(3,12)(4,10)(13,14)(17,21,23,35)(18,26,27,34)(19,31,32,46)(20,33,22,36)(24,39,40,48)(25,41,37,47)(28,42,38,45)(29,43,30,44)

Polycyclic presentation:

2^10|a^2=j,b^2=g,d^2=ij,e^2=j,f^2=i,g^2=ij,b^a=bf,c^a=chj,c^b=chi,d^b=df,d^c=dh,e^a=ei,e^b=eij,e^d=ei,f^a=fi,f^b=fh,f^c=fj,f^d=fi,g^a=ghi,g^c=gj,g^d=ghi,g^f=gj,h^b=hj

Conjugacy classes & character table

109 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318804
maximal normal subgroup of	#72318804

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.10);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 4, F.9*F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);