#6456840

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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

1522 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)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,25)(19,27,30,23)(20,31,33,24)(22,26,35,42)(28,38,47,45)(29,43,36,39)(32,40,46,48)(34,44,37,41);(1,3,5,4)(2,6,8,7)(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)(2,7,8,6)(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)(2,6,8,7)(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)(20,33)(24,31)(28,47)(32,46)(34,37)(38,45)(40,48)(41,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=fi,e^2=gi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,h^b=hi,e^c=ei,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, 6364036);