#6813670

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#42463	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1505 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,20)(21,28)(22,32)(23,27)(24,29)(25,31)(26,30);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,27)(21,22)(23,29)(25,32)(26,28)(30,31);(2,6)(5,8)(11,16)(13,15)(17,20,21,26,23,24,31,32)(18,19,28,30,27,29,25,22);(10,14)(13,15)(17,21,23,31)(18,25,27,28)(19,22,29,30)(20,26,24,32);(17,22,21,19,23,30,31,29)(18,26,28,24,27,32,25,20)

Polycyclic presentation:

2^9|b^2=g,c^2=fhi,d^2=i,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,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, 6720866);