#6503324

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500186
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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1688 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)(3,6)(4,5)(7,8)(17,18,22,27,23,28,21,35)(19,24,31,39,32,40,30,45)(20,26,33,42,34,41,36,25)(29,38,43,48,44,47,46,37);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,32)(18,24,28,40)(20,29,34,44)(21,30,22,31)(25,37,42,48)(26,38,41,47)(27,39,35,45)(33,43,36,46);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,29)(21,33)(22,36)(23,34)(24,37)(26,35)(27,41)(28,42)(30,43)(31,46)(32,44)(38,45)(39,47)(40,48);(11,16)(13,15)(17,20)(18,26)(19,29)(21,36)(22,33)(23,34)(24,38)(25,35)(27,42)(28,41)(30,46)(31,43)(32,44)(37,45)(39,48)(40,47);(10,14)(13,15)(17,21,23,22)(18,27,28,35)(19,30,32,31)(20,33,34,36)(24,39,40,45)(25,41,42,26)(29,43,44,46)(37,47,48,38)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,d^c=dh,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, 6410520);