#6754447

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	#39357	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1773 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:

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

Polycyclic presentation:

2^9|b^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, 6661643);