#6644482

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632082
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	#31036	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1375 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=hi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=efhi,c^b=ch,d^b=dfh,e^b=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6551678);