#6908560

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	#39334	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1814 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)(19,26)(20,22)(21,25)(23,31)(24,29)(27,30)(28,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(18,20)(24,30)(25,31)(26,32);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,21,27,22,28,23,29)(18,24,31,32,20,30,25,26);(2,6)(5,7)(11,16)(13,15)(17,20,22,18)(19,24,28,30)(21,25,23,31)(26,27,32,29);(10,14)(13,15)(17,21,22,23)(18,25,20,31)(19,27,28,29)(24,26,30,32)

Polycyclic presentation:

2^9|a^2=fi,c^2=gi,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,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, 6815756);