#6456842

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=f,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, 6364038);