#6757338

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	#39462	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1792 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6664534);