#6909318

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907909
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

1554 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=gi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh

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, 6816514);