#6580399

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

1532 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)(17,18,21,23)(19,24,30,36)(20,31,33,25)(22,26,34,37)(27,40,28,38)(29,45,46,39)(32,41,35,42)(43,47,44,48);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(18,23)(19,27)(20,32)(22,34)(24,38)(25,41)(28,30)(31,42)(33,35)(36,40)(39,45)(43,44);(10,14)(13,16)(17,19,21,30)(18,24,23,36)(20,29,33,46)(22,28,34,27)(25,39,31,45)(26,38,37,40)(32,43,35,44)(41,47,42,48);(1,4,5,3)(2,6,7,8)(19,28)(20,32)(22,34)(24,38)(25,42)(26,37)(27,30)(29,46)(31,41)(33,35)(36,40)(39,45);(17,20,21,33)(18,25,23,31)(19,29,30,46)(22,35,34,32)(24,39,36,45)(26,41,37,42)(27,43,28,44)(38,47,40,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=i,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,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, 6487595);