Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624842 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1209 conjugacy classes of subgroups.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 80 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18,23,28)(19,24,30,43)(20,33,35,60)(21,38,40,66)(22,41,39,27)(25,46,47,36)(26,49,50,34)(29,51,53,73)(31,58,56,79)(32,59,57,45)(37,65,64,62)(42,67,68,54)(44,71,70,52)(48,63,61,72)(55,78,77,75)(69,76,74,80);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19)(18,24)(20,29)(21,31)(22,32)(23,30)(25,42)(26,44)(27,45)(28,43)(33,51)(34,52)(35,53)(36,54)(37,55)(38,58)(39,57)(40,56)(41,59)(46,67)(47,68)(48,69)(49,71)(50,70)(60,73)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(72,80);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,34,63,22,37,40,46,23,35,49,72,39,64,21,36)(18,25,38,62,27,48,50,60,28,47,66,65,41,61,26,33)(19,29,52,76,32,55,56,67,30,53,71,80,57,77,31,54)(24,42,58,75,45,69,70,73,43,68,79,78,59,74,44,51);(3,12)(7,10)(11,13)(15,16)(20,35)(25,47)(29,53)(33,60)(36,46)(37,64)(42,68)(48,61)(51,73)(54,67)(55,77)(62,65)(63,72)(69,74)(75,78)(76,80);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(19,30)(24,43)(29,53)(31,56)(32,57)(42,68)(44,70)(45,59)(51,73)(52,71)(54,67)(55,77)(58,79)(69,74)(75,78)(76,80)Polycyclic presentation:
2^9|a^2=i,c^2=fgh,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532320);