Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624824 |
| 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 |
| prime radical | C23 | C22×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1143 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,23,28)(19,24,30,43)(20,33,37,62)(21,38,40,66)(22,41,39,27)(25,46,48,35)(26,49,50,34)(29,51,53,73)(31,58,56,79)(32,59,57,45)(36,65,64,61)(42,67,68,54)(44,71,70,52)(47,63,60,72)(55,78,77,75)(69,76,74,80);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,24)(20,29)(21,31)(22,32)(23,30)(25,42)(26,44)(27,45)(28,43)(33,51)(34,52)(35,54)(36,55)(37,53)(38,58)(39,57)(40,56)(41,59)(46,67)(47,69)(48,68)(49,71)(50,70)(60,74)(61,75)(62,73)(63,76)(64,77)(65,78)(66,79)(72,80);(17,20,34,63,22,36,40,46,23,37,49,72,39,64,21,35)(18,25,38,61,27,47,50,62,28,48,66,65,41,60,26,33)(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);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15);(2,8)(6,9)(7,16)(14,15)(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,b^2=f,c^2=gh,d^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=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532127);