#6631923

order	512
nilpotency class	3
isoclinism	isoclinic class of #6631112
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	#2856	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1590 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,c^b=ch,d^c=dfh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6539119);