#6914347

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	#39386	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1622 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6821543);