#6609355

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1366 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)(19,28)(20,33)(21,26)(22,27)(23,37)(24,30)(25,36)(29,43)(31,42)(32,39)(34,40)(35,48)(38,45)(41,46)(44,47);(1,3,5,4)(2,6,8,7)(9,10,12,13)(11,14,15,16)(17,19,22,30)(18,24,27,28)(20,29,35,44)(21,31,23,32)(25,38,40,41)(26,39,37,42)(33,47,48,43)(34,45,36,46);(1,3,5,4)(2,7,8,6)(10,13)(11,15)(17,20,21,34,22,35,23,36)(18,25,37,48,27,40,26,33)(19,29,31,45,30,44,32,46)(24,38,42,43,28,41,39,47);(1,4,5,3)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(19,30)(24,28)(29,44)(31,32)(38,41)(39,42)(43,47)(45,46);(10,13)(11,15)(17,21,22,23)(18,26,27,37)(19,31,30,32)(20,34,35,36)(24,39,28,42)(25,33,40,48)(29,45,44,46)(38,47,41,43)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6516551);