#6604493

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1353 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

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, 6511689);