#6579820

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1685 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,25)(21,27)(22,28)(23,26)(29,38)(30,40)(31,39)(32,41)(33,43)(34,45)(35,42)(36,46)(37,44)(47,48);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,32)(21,34)(22,36)(23,30)(25,41)(26,40)(27,45)(28,46)(29,33)(31,37)(35,47)(38,43)(39,44)(42,48);(10,14)(13,16)(17,20,23,21)(18,25,26,27)(19,29,30,47)(22,35,37,33)(24,38,40,48)(28,42,44,43)(31,34,36,32)(39,45,46,41);(1,3,4,7)(2,6,5,8)(18,26)(19,30)(20,33)(21,35)(22,37)(25,42)(27,43)(29,34)(32,47)(38,41)(39,46)(45,48);(17,21,23,20)(18,27,26,25)(19,29,30,47)(22,33,37,35)(24,38,40,48)(28,43,44,42)(31,34,36,32)(39,45,46,41)

Polycyclic presentation:

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

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, 6487016);