#6587699

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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

1542 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)(19,28)(20,25)(21,24)(22,26)(23,27)(29,47)(30,40)(31,39)(32,42)(33,43)(34,44)(35,38)(36,45)(37,46)(41,48);(2,6)(7,8)(17,19,22,21)(18,24,26,28)(20,32,33,48)(23,30,36,31)(25,41,43,42)(27,39,45,40)(29,37,35,34)(38,46,47,44);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,33)(18,25,26,43)(19,29,21,35)(23,37,36,34)(24,38,28,47)(27,46,45,44)(30,32,31,48)(39,41,40,42);(9,11,12,15)(10,14,13,16)(17,21,22,19)(18,24,26,28)(20,32,33,48)(23,31,36,30)(25,42,43,41)(27,39,45,40)(29,37,35,34)(38,44,47,46);(1,4,5,3)(2,8,6,7)(19,30)(20,33)(21,31)(23,36)(24,39)(25,43)(27,45)(28,40)(29,48)(32,35)(38,42)(41,47)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=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, 6494895);