#6763226

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

1854 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)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,28,30,24)(20,29,27,23)(21,31,32,25);(1,2,4,5)(3,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,25)(21,23)(22,30)(26,28)(27,31)(29,32);(10,14)(11,15)(17,20,23,18,22,27,29,26)(19,25,21,24,30,31,32,28);(1,3,4,7)(2,6,5,8)(17,21)(18,25)(19,29)(20,28)(22,32)(23,30)(24,27)(26,31);(2,5)(6,8)(11,15)(13,16)(19,30)(21,32)(24,28)(25,31)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gh,d^2=fi,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, 6670422);