#6582686

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1371 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=fi,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6489882);