#6509431

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499804
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×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1484 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)(9,10,13,12)(11,15,16,14)(17,18)(19,28)(20,33)(21,26)(22,27)(23,37)(24,30)(25,36)(29,45)(31,44)(32,40)(34,48)(35,43)(38,42)(39,47)(41,46);(2,5)(6,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,24)(20,34)(21,31)(22,30)(23,32)(25,41)(26,40)(27,28)(29,35)(33,45)(36,47)(37,44)(38,46)(39,42)(43,48);(10,12)(11,16)(17,20,22,35)(18,25,27,42)(19,29,30,34)(21,36,23,38)(24,39,28,41)(26,43,37,33)(31,46,32,47)(40,45,44,48);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,15,16,14)(19,30)(20,35)(24,28)(25,42)(31,32)(33,43)(36,38)(40,44);(11,16)(14,15)(17,21,22,23)(18,26,27,37)(19,31,30,32)(20,36,35,38)(24,40,28,44)(25,43,42,33)(29,46,34,47)(39,45,41,48)

Polycyclic presentation:

2^9|a^2=hi,c^2=i,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=ehi,d^c=dh,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6416627);