#6581124

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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1380 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)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,29,32,24)(20,25,23,28)(21,26,35,41)(30,43,45,37)(31,39,33,38)(34,40,36,42)(44,48,46,47);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,33)(21,30)(22,32)(23,31)(25,39)(26,37)(27,29)(28,38)(34,44)(35,45)(36,46)(40,47)(41,43)(42,48);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,20)(18,25)(22,23)(27,28)(30,44)(31,33)(34,36)(37,47)(38,39)(40,42)(43,48)(45,46);(1,4,5,3)(2,8,6,7)(17,21,22,35)(18,26,27,41)(19,30,32,45)(20,34,23,36)(24,37,29,43)(25,40,28,42)(31,46,33,44)(38,48,39,47);(19,31)(20,23)(21,34)(24,38)(25,28)(26,40)(29,39)(30,45)(32,33)(35,36)(37,43)(41,42)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fi,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,e^c=ei,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, 6488320);