#6586225

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1459 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,15,16,13)(17,18,22,27)(19,29,31,24)(20,25,34,42)(21,26,35,43)(23,28,37,45)(30,44,36,39)(32,47,48,40)(33,46,38,41);(1,3,5,4)(2,6,7,8)(17,19,22,31)(18,24,27,29)(20,32,34,48)(21,36,35,30)(23,38,37,33)(25,40,42,47)(26,44,43,39)(28,46,45,41);(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,30)(21,23)(22,34)(24,39)(26,28)(27,42)(29,44)(31,36)(32,38)(33,48)(35,37)(40,46)(41,47)(43,45);(1,4,5,3)(2,6,7,8)(19,31)(20,21)(23,37)(24,29)(25,26)(28,45)(30,32)(34,35)(36,48)(39,40)(42,43)(44,47);(17,21,22,35)(18,26,27,43)(19,32,31,48)(20,23,34,37)(24,40,29,47)(25,28,42,45)(30,38,36,33)(39,46,44,41)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=g,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=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, 6493421);