#6635547

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632088
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37448	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1819 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)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18,23,32,22,26,21,25)(19,27,28,24,30,31,29,20);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,15)(13,14);(3,7)(5,8)(9,11,13,12)(10,15,14,16)(17,19,22,30)(18,24,26,20)(21,28,23,29)(25,31,32,27);(3,7)(5,8)(9,12,13,11)(10,15,14,16)(17,20)(18,19)(21,31)(22,24)(23,27)(25,29)(26,30)(28,32);(2,6)(5,8)(17,21,22,23)(18,25,26,32)(19,28,30,29)(20,27,24,31)

Polycyclic presentation:

2^9|a^2=fgi,c^2=fi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh,g^c=gi,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, 6542743);