#6629571

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

1778 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,h^a=hi,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, 6536767);