#6502066

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1650 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)(17,18,22,28)(19,29,33,24)(20,26,36,43)(21,27,23,38)(25,37,44,35)(30,40,31,45)(32,46,34,41)(39,47,42,48);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,31)(21,32)(22,33)(23,34)(25,42)(26,40)(27,41)(28,29)(30,36)(35,48)(37,47)(38,46)(39,44)(43,45);(9,11)(10,13)(12,16)(14,15)(17,20,22,36)(18,25,28,44)(19,30,33,31)(21,35,23,37)(24,39,29,42)(26,27,43,38)(32,47,34,48)(40,46,45,41);(11,16)(13,15)(17,20)(18,26)(19,31)(21,37)(22,36)(23,35)(24,40)(25,38)(27,44)(28,43)(29,45)(30,33)(32,47)(34,48)(39,41)(42,46);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,22,23)(18,27,28,38)(19,32,33,34)(20,35,36,37)(24,41,29,46)(25,43,44,26)(30,47,31,48)(39,40,42,45)

Polycyclic presentation:

2^9|a^2=i,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,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, 6409262);