#6535558

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1467 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)(3,6)(4,5)(7,8)(10,13)(14,16)(17,18)(19,28)(20,33)(21,27)(22,36)(23,26)(24,31)(25,34)(29,38)(30,39)(32,42)(35,41)(37,40)(43,44)(45,47)(46,48);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,32)(18,24,26,39)(20,29,35,45)(21,31,22,30)(25,38,40,47)(27,42,36,28)(33,43,41,48)(34,46,37,44);(10,13)(14,16)(17,20,23,35)(18,25,26,40)(19,29,32,45)(21,34,22,37)(24,38,39,47)(27,41,36,33)(28,43,42,48)(30,44,31,46);(1,3,4,7)(2,6,5,8)(11,15)(14,16)(18,26)(24,39)(25,40)(27,36)(28,42)(33,41)(38,47)(43,48);(9,11,12,15)(10,14,13,16)(17,21,23,22)(18,27,26,36)(19,30,32,31)(20,34,35,37)(24,28,39,42)(25,41,40,33)(29,44,45,46)(38,43,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6442754);