#6761908

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1868 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,23,24,32)(20,28,31,27)(21,29,30,25);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,29)(21,27)(22,24)(25,31)(26,32)(28,30);(10,14)(11,15)(17,20,30,32,22,31,21,23)(18,24,29,27,26,19,25,28);(2,6)(5,7)(17,19)(18,23)(20,29)(21,27)(22,24)(25,31)(26,32)(28,30);(3,8)(5,7)(11,15)(13,16)(19,24)(20,31)(23,32)(27,28)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6669104);