#6580375

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1550 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)(19,27)(20,32)(21,23)(22,26)(24,31)(25,34)(28,38)(29,40)(30,39)(33,41)(35,37)(36,42)(43,46)(44,48)(45,47);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(18,23)(19,28)(20,33)(22,35)(24,38)(25,41)(27,40)(29,31)(32,42)(34,36)(39,43)(44,45);(10,14)(13,16)(17,19,21,31)(18,24,23,27)(20,30,34,46)(22,29,35,28)(25,39,32,43)(26,38,37,40)(33,44,36,45)(41,47,42,48);(1,4,5,3)(2,6,7,8)(19,29)(20,33)(22,35)(24,38)(25,42)(26,37)(27,40)(28,31)(30,46)(32,41)(34,36)(39,43);(17,20,21,34)(18,25,23,32)(19,30,31,46)(22,36,35,33)(24,39,27,43)(26,41,37,42)(28,44,29,45)(38,47,40,48)

Polycyclic presentation:

2^9|b^2=fg,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, 6487571);