#6911368

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907977
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1771 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fgi,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=efhi,d^c=dfh,e^c=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, 6818564);