#6585853

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1398 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6493049);