#6909308

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

1563 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)(17,18,23,26)(19,24,28,29)(20,27,30,31)(21,25,22,32);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,25)(21,31)(22,27)(23,28)(26,29)(30,32);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,20,22,29,23,30,21,24)(18,19,25,27,26,28,32,31);(2,6)(5,7)(11,16)(13,15)(17,21,23,22)(18,25,26,32)(19,27,28,31)(20,24,30,29);(10,14)(13,15)(17,21,23,22)(18,25,26,32)(19,27,28,31)(20,24,30,29)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=g,d^2=i,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, 6816504);