#6459283

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1474 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)(17,18,22,26,23,27,36,42)(19,28,21,34,30,41,35,24)(20,25,33,40,32,39,43,47)(29,38,31,45,44,48,46,37);(2,6)(7,8)(17,19,23,30)(18,24,27,34)(20,31,32,46)(21,22,35,36)(25,38,39,48)(26,41,42,28)(29,43,44,33)(37,47,45,40);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,33,23,32,36,43)(18,25,26,40,27,39,42,47)(19,29,21,31,30,44,35,46)(24,37,28,38,34,45,41,48);(1,4,5,3)(2,8,6,7)(17,21,22,30,23,35,36,19)(18,24,26,28,27,34,42,41)(20,31,33,44,32,46,43,29)(25,37,40,38,39,45,47,48);(9,11,12,15)(10,14,13,16)(20,32)(25,39)(29,44)(31,46)(33,43)(37,45)(38,48)(40,47)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,d^2=f,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366479);