#232391

order	512
nilpotency class	3
isoclinism	isoclinic class of #232373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1331 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=hi,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=gi,c^b=cg,d^b=dhi,e^b=eh,g^b=gh,e^c=ei,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139587);