#6447291

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1712 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:

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

Polycyclic presentation:

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

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, 6354487);