#6630405

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1734 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)(9,10,12,14)(11,15,16,13)(17,18,21,24)(19,23,28,36)(20,30,31,40)(22,34,33,26)(25,35,39,32)(27,41,42,48)(29,45,44,38)(37,46,47,43);(2,5)(6,8)(17,19)(18,23)(20,27)(21,28)(22,29)(24,36)(25,37)(26,38)(30,41)(31,42)(32,43)(33,44)(34,45)(35,46)(39,47)(40,48);(9,11,12,16)(10,13,14,15)(18,24)(23,36)(25,39)(26,34)(30,40)(37,47)(38,45)(41,48);(17,20,21,31)(18,25,24,39)(19,27,28,42)(22,35,33,32)(23,37,36,47)(26,40,34,30)(29,46,44,43)(38,48,45,41);(1,3,4,7)(2,6,5,8)(19,28)(23,36)(27,42)(29,44)(37,47)(38,45)(41,48)(43,46)

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dh,h^a=hi,e^b=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, 6537601);