#6843144

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#26968	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1451 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18,23,22)(19,24,27,32)(20,25,29,28)(21,30,31,26);(2,5)(3,7)(17,19)(18,24)(20,21)(22,32)(23,27)(25,30)(26,28)(29,31);(10,14)(15,16)(17,20,22,28,23,29,18,25)(19,26,24,21,27,30,32,31);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,18,23,22)(19,24,27,32)(20,28,29,25)(21,26,31,30);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,21,18,26,23,31,22,30)(19,25,32,29,27,28,24,20)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=gi,e^2=fghi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=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, 6750340);