#6508344

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1564 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,8)(4,6)(5,7)(17,18)(19,28)(20,26)(21,27)(22,24)(23,37)(25,32)(29,45)(30,38)(31,44)(33,48)(34,42)(35,39)(36,43)(40,46)(41,47);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(18,24)(25,38)(26,39)(27,37)(28,44)(40,41)(42,43)(45,48);(9,11)(10,13)(12,16)(14,15)(17,19,23,32,22,31,21,30)(18,25,27,28,24,38,37,44)(20,33,34,47,35,29,36,46)(26,41,43,48,39,40,42,45);(11,16)(13,15)(17,20)(18,26)(19,29)(21,36)(22,35)(23,34)(24,39)(25,40)(27,43)(28,45)(30,47)(31,33)(32,46)(37,42)(38,41)(44,48);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,22,23)(18,27,24,37)(19,30,31,32)(20,34,35,36)(25,28,38,44)(26,42,39,43)(29,46,33,47)(40,48,41,45)

Polycyclic presentation:

2^9|b^2=fi,c^2=g,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,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, 6415540);