#6602135

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1496 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)(4,6)(5,8)(17,18,22,27)(19,28,31,24)(20,32,34,44)(21,26,23,37)(25,38,43,35)(29,45,33,48)(30,41,36,40)(39,47,42,46);(1,3,5,4)(2,6,8,7)(9,10,12,13)(11,14,15,16)(17,19,22,31)(18,24,27,28)(20,33,34,29)(21,36,23,30)(25,42,43,39)(26,41,37,40)(32,45,44,48)(35,46,38,47);(1,3,5,4)(2,7,8,6)(10,13)(11,15)(17,20,22,34)(18,25,27,43)(19,29,31,33)(21,35,23,38)(24,39,28,42)(26,44,37,32)(30,46,36,47)(40,48,41,45);(1,4,5,3)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(20,34)(25,43)(29,33)(32,44)(35,38)(39,42)(45,48)(46,47);(10,13)(11,15)(17,21,22,23)(18,26,27,37)(19,30,31,36)(20,35,34,38)(24,40,28,41)(25,44,43,32)(29,46,33,47)(39,48,42,45)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=ehi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6509331);