#6580543

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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1537 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)(3,8)(4,6)(5,7)(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)(2,6,7,8)(9,11,12,15)(10,13,14,16)(18,23)(19,27)(20,30)(22,33)(24,38)(25,41)(29,34)(31,32)(35,39)(36,42)(40,46)(43,45);(10,14)(13,16)(17,19,21,29)(18,24,23,35)(20,28,32,44)(22,34,33,27)(25,40,36,46)(26,38,37,39)(30,43,31,45)(41,47,42,48);(1,4,5,3)(2,6,7,8)(19,27)(20,31)(22,33)(24,39)(25,41)(26,37)(28,44)(29,34)(30,32)(35,38)(36,42)(40,46);(17,20,21,32)(18,25,23,36)(19,28,29,44)(22,31,33,30)(24,40,35,46)(26,41,37,42)(27,43,34,45)(38,47,39,48)

Polycyclic presentation:

2^9|b^2=fgi,c^2=i,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=ehi,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, 6487739);