#6909316

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	#37670	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)(17,18,23,19)(20,28,31,32)(21,25,22,27)(24,29,26,30);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,18)(20,26,31,24)(21,25,22,27)(28,30,32,29);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,20,22,30,23,31,21,29)(18,24,25,28,19,26,27,32);(2,6)(5,7)(11,16)(13,15)(17,21,23,22)(18,25,19,27)(20,29,31,30)(24,28,26,32);(10,14)(13,15)(17,21,23,22)(18,25,19,27)(20,29,31,30)(24,28,26,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,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, 6816512);