#6452875

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1645 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)(11,16)(12,14)(13,15)(17,18,22,27)(19,28,32,24)(20,33,35,25)(21,38,23,26)(29,39,47,45)(30,41,31,40)(34,42,46,48)(36,43,37,44);(2,5)(6,8)(17,19,22,32)(18,24,27,28)(20,34,35,46)(21,31,23,30)(25,42,33,48)(26,40,38,41)(29,36,47,37)(39,44,45,43);(9,11,13,12)(10,14,15,16)(17,20,22,35)(18,25,27,33)(19,29,32,47)(21,37,23,36)(24,39,28,45)(26,44,38,43)(30,34,31,46)(40,42,41,48);(9,12,13,11)(10,14,15,16)(19,30)(20,35)(21,23)(24,40)(25,33)(26,38)(28,41)(29,46)(31,32)(34,47)(39,48)(42,45);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,26,27,38)(19,31,32,30)(20,36,35,37)(24,40,28,41)(25,43,33,44)(29,34,47,46)(39,48,45,42)

Polycyclic presentation:

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

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, 6360071);