#6587765

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	C₂×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1532 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,27)(20,25)(21,24)(22,26)(23,30)(28,36)(29,43)(31,42)(32,41)(33,40)(34,38)(35,39)(37,45)(44,47)(46,48);(2,6)(7,8)(17,19,21,30)(18,23,24,27)(20,29,34,45)(22,28,35,31)(25,37,38,43)(26,42,39,36)(32,46,33,44)(40,48,41,47);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(18,24)(19,28)(20,32)(22,35)(23,36)(25,40)(27,42)(29,45)(30,31)(33,34)(38,41)(47,48);(9,11,12,15)(10,14,13,16)(17,20,21,34)(18,25,24,38)(19,29,30,45)(22,33,35,32)(23,37,27,43)(26,40,39,41)(28,44,31,46)(36,47,42,48);(1,4,5,3)(2,8,6,7)(18,24)(19,28)(20,33)(22,35)(23,36)(25,41)(27,42)(30,31)(32,34)(37,43)(38,40)(44,46)

Polycyclic presentation:

2^9|b^2=i,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, 6494961);