#6582895

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1348 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)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,24)(19,27,31,23)(20,25,34,39)(22,26,36,40)(28,37,30,44)(29,41,32,38)(33,43,35,42)(45,47,46,48);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,19,21,31)(18,23,24,27)(20,32,34,29)(22,28,36,30)(25,41,39,38)(26,44,40,37)(33,46,35,45)(42,48,43,47);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(18,24)(19,28)(20,33)(22,36)(23,37)(25,42)(27,44)(30,31)(34,35)(38,41)(39,43)(45,46);(1,3,5,4)(2,7,8,6)(17,20)(18,25)(19,29)(21,34)(22,33)(23,38)(24,39)(26,43)(27,41)(28,45)(30,46)(31,32)(35,36)(37,47)(40,42)(44,48);(1,4,5,3)(2,6,8,7)(19,30)(20,33)(22,36)(23,37)(25,43)(26,40)(27,44)(28,31)(29,32)(34,35)(38,41)(39,42)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=fi,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,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, 6490091);