#6528623

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500456
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1365 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=g,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,d^c=dhi,e^d=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, 6435819);