#6456244

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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1564 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)(11,14)(12,13)(15,16)(17,18)(19,23)(20,30)(21,25)(22,26)(24,32)(27,45)(28,38)(29,39)(31,48)(33,44)(34,42)(35,43)(36,41)(37,47)(40,46);(2,6)(7,8)(17,19,21,29)(18,23,25,39)(20,31,32,46)(22,35,34,28)(24,40,30,48)(26,43,42,38)(27,36,47,33)(37,44,45,41);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,21,32)(18,24,25,30)(19,27,29,47)(22,36,34,33)(23,37,39,45)(26,44,42,41)(28,46,35,31)(38,48,43,40);(1,4,5,3)(2,8,6,7)(19,28)(22,34)(23,38)(26,42)(27,46)(29,35)(31,47)(33,36)(37,48)(39,43)(40,45)(41,44);(9,11,12,15)(10,14,13,16)(18,25)(19,29)(24,30)(26,42)(27,47)(28,35)(31,46)(41,44)

Polycyclic presentation:

2^9|b^2=i,c^2=fi,d^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,h^c=hi,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, 6363440);