#6455055

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1478 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,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, 6362251);