#6874881

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	#30985	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1521 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,d^2=fh,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,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, 6782077);