#6581189

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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 ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1363 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)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,23,29,38)(20,30,33,24)(21,25,34,41)(27,36,35,42)(28,44,45,37)(31,40,32,39)(43,48,46,47);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(19,27)(20,31)(21,34)(23,36)(24,39)(25,41)(29,35)(30,40)(32,33)(38,42)(43,46)(47,48);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,19,22,29)(18,23,26,38)(20,28,33,45)(21,35,34,27)(24,37,30,44)(25,42,41,36)(31,43,32,46)(39,47,40,48);(1,4,5,3)(2,8,6,7)(19,27)(20,32)(21,34)(23,36)(24,40)(25,41)(28,45)(29,35)(30,39)(31,33)(37,44)(38,42);(17,20,22,33)(18,24,26,30)(19,28,29,45)(21,32,34,31)(23,37,38,44)(25,40,41,39)(27,43,35,46)(36,47,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=ehi,h^d=hi,h^e=hi

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, 6488385);