#6651062

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632088
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	#35607	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfhi,e^b=ei,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6558258);