#6580743

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

1534 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)(17,18,22,27)(19,29,32,24)(20,25,36,43)(21,37,38,26)(23,28,35,45)(30,44,48,40)(31,41,34,47)(33,46,39,42);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19,22,32)(18,24,27,29)(20,34,36,31)(21,30,38,48)(23,39,35,33)(25,41,43,47)(26,44,37,40)(28,46,45,42);(10,14)(13,16)(17,20,22,36)(18,25,27,43)(19,30,32,48)(21,23,38,35)(24,40,29,44)(26,45,37,28)(31,39,34,33)(41,42,47,46);(1,4,5,3)(2,6,7,8)(20,21)(23,35)(25,37)(26,43)(28,45)(30,34)(31,48)(33,39)(36,38)(40,41)(42,46)(44,47);(17,21,22,38)(18,26,27,37)(19,31,32,34)(20,35,36,23)(24,41,29,47)(25,28,43,45)(30,33,48,39)(40,46,44,42)

Polycyclic presentation:

2^9|a^2=i,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=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, 6487939);