#6763575

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1736 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18,21,22)(19,26,27,32)(20,24,30,25)(23,31,28,29);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(18,22)(19,27)(24,25)(29,31);(2,6)(3,7)(10,13)(14,16)(17,19,20,29,21,27,30,31)(18,23,25,26,22,28,24,32);(9,11,12,15)(10,14,13,16)(17,18)(19,28)(20,25)(21,22)(23,27)(24,30)(26,31)(29,32);(3,7)(5,8)(17,20,21,30)(18,24,22,25)(19,29,27,31)(23,32,28,26)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6670771);