#6455185

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1527 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,27)(19,24,32,41)(20,33,36,25)(21,26,23,28)(29,45,47,38)(30,39,31,40)(34,48,46,42)(35,44,37,43);(2,5)(6,8)(17,19,22,32)(18,24,27,41)(20,34,36,46)(21,31,23,30)(25,42,33,48)(26,40,28,39)(29,37,47,35)(38,44,45,43);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20)(18,25)(19,29)(21,35)(22,36)(23,37)(24,38)(26,43)(27,33)(28,44)(30,34)(31,46)(32,47)(39,42)(40,48)(41,45);(19,30)(21,23)(24,39)(26,28)(29,34)(31,32)(35,37)(38,42)(40,41)(43,44)(45,48)(46,47);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,26,27,28)(19,31,32,30)(20,35,36,37)(24,40,41,39)(25,43,33,44)(29,46,47,34)(38,48,45,42)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,h^c=hi,e^d=ei,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, 6362381);