#6629923

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628851
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1785 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)(11,13)(12,14)(15,16)(17,18)(19,30)(20,33)(21,25)(22,28)(23,29)(24,39)(26,36)(27,37)(31,40)(32,41)(34,42)(35,45)(38,44)(43,47)(46,48);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,23,21)(18,25,29,30)(20,34,36,31)(22,37,24,32)(26,42,33,40)(27,28,41,39)(35,46,38,47)(43,44,48,45);(10,14)(13,16)(17,20)(18,26)(19,31)(21,34)(22,38)(23,36)(24,35)(25,40)(27,43)(28,45)(29,33)(30,42)(32,47)(37,46)(39,44)(41,48);(17,21,23,19)(18,27,29,41)(20,34,36,31)(22,37,24,32)(25,28,30,39)(26,43,33,48)(35,47,38,46)(40,45,42,44);(1,3,4,7)(2,6,5,8)(17,22,23,24)(18,28,29,39)(19,32,21,37)(20,35,36,38)(25,27,30,41)(26,44,33,45)(31,46,34,47)(40,48,42,43)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6537119);