#6581535

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1514 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)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,25)(21,23)(22,26)(27,39)(28,40)(29,35)(30,42)(31,41)(32,36)(33,37)(34,38)(43,48)(44,46)(45,47);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(18,23)(19,27)(20,30)(22,33)(24,38)(25,41)(28,44)(29,34)(31,32)(35,39)(36,42)(47,48);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,19)(18,24)(20,28)(21,29)(22,34)(23,35)(25,40)(26,38)(27,33)(30,45)(31,43)(32,44)(36,46)(37,39)(41,48)(42,47);(1,4,5,3)(2,6,8,7)(19,27)(20,31)(22,33)(24,39)(25,41)(26,37)(29,34)(30,32)(35,38)(36,42)(43,45)(47,48);(17,20,21,32)(18,25,23,36)(19,28,29,44)(22,30,33,31)(24,40,35,46)(26,42,37,41)(27,43,34,45)(38,47,39,48)

Polycyclic presentation:

2^9|b^2=fgi,c^2=fi,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6488731);