#6667398

order	512
nilpotency class	3
isoclinism	isoclinic class of #6660402
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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1443 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)(3,8)(4,6)(5,7)(11,15)(14,16)(17,18)(19,23)(20,25)(21,24)(22,26)(27,30)(28,31)(29,32);(1,3,6,5)(2,7,4,8)(10,13)(14,16)(17,19,22,21)(18,23,26,24)(20,28,29,27)(25,31,32,30);(1,4,6,2)(3,8,5,7)(9,10)(11,14)(12,13)(15,16)(18,24)(19,21)(20,29)(23,26)(25,30)(31,32);(2,4)(7,8)(10,13)(14,16)(17,20)(18,25)(19,27)(21,28)(22,29)(23,30)(24,31)(26,32);(1,5,6,3)(2,8,4,7)(9,11,12,15)(10,14,13,16)(20,29)(25,32)(27,28)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6574594);