#6587841

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1507 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)(17,18,22,26)(19,28,21,24)(20,32,34,25)(23,27,37,44)(29,39,36,47)(30,40,31,45)(33,41,48,42)(35,46,38,43);(2,6)(7,8)(17,19,22,21)(18,24,26,28)(20,33,34,48)(23,30,37,31)(25,41,32,42)(27,45,44,40)(29,38,36,35)(39,46,47,43);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,34)(18,25,26,32)(19,29,21,36)(23,38,37,35)(24,39,28,47)(27,46,44,43)(30,33,31,48)(40,42,45,41);(9,11,12,15)(10,14,13,16)(17,21,22,19)(18,24,26,28)(20,33,34,48)(23,31,37,30)(25,42,32,41)(27,45,44,40)(29,38,36,35)(39,43,47,46);(1,4,5,3)(2,8,6,7)(18,26)(19,30)(20,34)(21,31)(23,37)(24,40)(28,45)(29,48)(33,36)(39,41)(42,47)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,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,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6495037);