#6486321

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470161
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

1381 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)(25,26,31,33,30,28,29,34)(27,35,37,32,39,40,38,36);(2,5)(6,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27)(26,32)(28,36)(29,37)(30,39)(31,38)(33,35)(34,40);(11,17)(14,21)(18,19)(22,23)(25,28,30,26)(27,36,39,32)(29,34,31,33)(35,37,40,38);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,29,30,31)(26,33,28,34)(27,37,39,38)(32,35,36,40);(1,3,4,7)(2,6,5,8)(10,16)(14,21)(15,24)(22,23)(25,29,30,31)(26,33,28,34)(27,38,39,37)(32,40,36,35)

Polycyclic presentation:

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

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, 6393517);