#6499415

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470368
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
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1466 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,7)(4,5)(6,8)(9,10)(11,14)(12,19)(13,15)(16,20)(17,21)(18,24)(22,23)(25,26)(27,29)(28,36)(30,35)(31,33)(32,37)(34,39)(38,40);(2,5)(6,7)(9,11,13,17)(10,14,15,21)(12,18,20,23)(16,22,19,24)(25,27,29,26,31,35,30,33)(28,34,38,32,37,40,39,36);(1,3,4,8)(2,6,5,7)(10,15)(11,17)(16,19)(18,23)(25,28,31,37)(26,32,33,36)(27,34,35,40)(29,38,30,39);(11,17)(14,21)(18,23)(22,24)(28,37)(32,36)(34,40)(38,39);(9,12,13,20)(10,16,15,19)(11,18,17,23)(14,22,21,24)(25,29,31,30)(26,27,33,35)(28,38,37,39)(32,34,36,40)

Polycyclic presentation:

2^9|b^2=fg,c^2=gh,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di

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, 6406611);