#72319446

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	C₂×C₄	C₂²×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂²	C₂³×#102
prime radical	C₂×#61431	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ C₂×C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5067 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=fgij,f^2=h,g^2=i,h^2=j,i^2=j,b^a=bj,c^a=cj,c^b=cg,d^b=dhi,d^c=dhi,e^b=ehi,e^d=ehij,f^c=fi,g^b=gj,g^c=gi,h^c=hj,i^c=ij

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319510, #72319621, #72319749, …
maximal normal subgroup of	#72319510, #72319621, #72319749, …

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