#6910805

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

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=f,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, 6818001);