#6793787

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#24382	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1617 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)(17,18,21,26,22,27,29,24)(19,23,30,31,28,32,20,25);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,13)(14,16)(17,19)(18,23)(20,29)(21,30)(22,28)(24,25)(26,31)(27,32);(10,13)(14,15)(18,24)(19,28)(21,29)(23,31)(25,32)(26,27);(2,6)(5,7)(17,20,22,30)(18,25,27,31)(19,29,28,21)(23,24,32,26);(3,8)(5,7)(9,11,12,16)(10,14,13,15)(19,28)(20,30)(23,32)(25,31)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6700983);