#6651043

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	#2416	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1418 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)(26,30)(27,33)(29,36)(31,39)(34,40)(35,37);(3,4)(6,7)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,26,28,30)(27,34,35,31)(29,32,36,38)(33,39,37,40);(1,3,5,4)(2,6,8,7)(25,27,28,35)(26,31,30,34)(29,37,36,33)(32,39,38,40);(1,4,5,3)(2,7,8,6)(10,15)(11,18)(16,20)(19,24)(25,26)(27,31)(28,30)(29,32)(33,40)(34,35)(36,38)(37,39);(2,8)(6,7)(9,12,13,21)(10,16,15,20)(11,19,18,24)(14,22,17,23)(26,30)(27,35)(32,38)(33,37)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=hi,e^2=fi,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, 6558239);