#6502321

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1644 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)(19,24)(20,26)(21,27)(22,37)(23,28)(25,33)(29,46)(30,39)(31,40)(32,42)(34,44)(35,41)(36,45)(38,48)(43,47);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,32)(18,24,28,42)(20,30,34,29)(21,35,22,31)(25,43,45,38)(26,39,44,46)(27,41,37,40)(33,47,36,48);(9,11)(10,13)(12,16)(14,15)(17,20,23,34)(18,25,28,45)(19,29,32,30)(21,33,22,36)(24,38,42,43)(26,27,44,37)(31,47,35,48)(39,40,46,41);(11,16)(13,15)(17,20)(18,26)(19,30)(21,36)(22,33)(23,34)(24,39)(25,37)(27,45)(28,44)(29,32)(31,47)(35,48)(38,41)(40,43)(42,46);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,23,22)(18,27,28,37)(19,31,32,35)(20,33,34,36)(24,40,42,41)(25,44,45,26)(29,47,30,48)(38,46,43,39)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,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, 6409517);