#6590723

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1499 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)(19,27)(20,24)(21,25)(22,26)(23,30)(28,38)(29,43)(31,36)(32,39)(33,42)(34,40)(35,41)(37,45)(44,48)(46,47);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,15,16)(17,19,21,30)(18,23,25,27)(20,29,34,45)(22,28,35,31)(24,37,40,43)(26,36,41,38)(32,46,33,44)(39,48,42,47);(1,3,5,4)(2,7,8,6)(10,15)(14,16)(19,28)(20,32)(22,35)(23,36)(24,39)(26,41)(27,38)(29,45)(30,31)(33,34)(37,43)(40,42);(9,12,13,11)(10,16,15,14)(17,20)(18,24)(19,29)(21,34)(22,33)(23,37)(25,40)(26,42)(27,43)(28,44)(30,45)(31,46)(32,35)(36,47)(38,48)(39,41);(1,4,5,3)(2,6,8,7)(18,25)(19,28)(20,33)(22,35)(23,38)(24,39)(27,36)(30,31)(32,34)(37,43)(40,42)(44,46)

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=fi,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,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, 6497919);