#72284994

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁵
prime radical	#34963	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

4634 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|b^2=j,c^2=gi,e^2=j,f^2=i,c^a=cg,c^b=cf,d^a=dhi,d^c=dhij,e^a=ei,e^b=ej,e^c=ei,f^a=fh,f^b=fi,f^c=fhij,g^b=ghj,g^d=gj,g^f=gj,h^c=hj

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285576
maximal normal subgroup of	#72285576

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, 2, F.10);;
gap> SetPower(coll, 3, F.7*F.9);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);