#6756752

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	#39416	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1573 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)(3,6)(4,8)(5,7)(18,20)(19,26)(21,23)(24,31)(27,32)(28,30);(1,3,6,2)(4,8,5,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,22)(21,29)(23,25)(26,32)(27,28)(30,31);(9,11,12,16)(10,13,14,15)(17,19,22,28)(18,24,20,27)(21,26,23,30)(25,31,29,32);(1,4,6,5)(2,7,3,8)(17,20,22,18)(19,27,28,24)(21,25,23,29)(26,31,30,32);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(17,21,22,23)(18,25,20,29)(19,26,28,30)(24,31,27,32)

Polycyclic presentation:

2^9|b^2=fi,c^2=hi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6663948);