#6754167

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1459 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=h,d^2=fi,e^2=i,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, 6661363);