#6486331

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1361 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)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,30,27,31,34,29,33)(28,32,38,35,36,39,37,40);(2,5)(6,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,31,33)(26,29,34,30)(28,35,36,40)(32,37,39,38);(11,17)(14,21)(18,19)(22,23)(25,26)(27,29)(28,32)(30,33)(31,34)(35,37)(36,39)(38,40);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,28)(26,32)(27,35)(29,37)(30,38)(31,36)(33,40)(34,39);(1,3,4,7)(2,6,5,8)(10,16)(14,21)(15,24)(22,23)(28,36)(32,39)(35,40)(37,38)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=h,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6393527);