#6843145

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	#26634	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1430 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)(17,18,22,23)(19,20,26,28)(21,29,31,24)(25,30,32,27);(2,5)(3,7)(17,19)(18,20)(21,27)(22,26)(23,28)(24,32)(25,29)(30,31);(10,14)(15,16)(17,20,18,19,22,28,23,26)(21,30,24,32,31,27,29,25);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,18,22,23)(19,20,26,28)(21,24,31,29)(25,27,32,30);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,21,18,24,22,31,23,29)(19,25,28,30,26,32,20,27)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=g,e^2=fgh,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, 6750341);