#6907789

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
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	#30513	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1371 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)(3,6)(4,8)(5,7)(17,18,21,26,22,28,23,27)(19,20,30,32,25,24,29,31);(1,2,6,3)(4,7,5,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,25)(18,24,28,20)(21,30,23,29)(26,31,27,32);(1,3,6,2)(4,7,5,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,28)(21,31)(22,24)(23,32)(26,30)(27,29);(1,4,6,5)(2,7,3,8)(11,16)(13,15)(17,21,22,23)(18,26,28,27)(19,29,25,30)(20,31,24,32);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(17,21,22,23)(18,27,28,26)(19,30,25,29)(20,31,24,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6814985);