#6456246

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

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=fi,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, 6363442);