#6910802

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	#37748	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1630 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6817998);