#6501595

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=fi,d^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, 6408791);