Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219498 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,3)(5,6,11,16,12,18,10,17)(7,19,23,38,24,37,22,13)(8,25,28,42,30,41,29,14)(9,31,33,44,34,43,32,15)(20,47,49,45,50,35,26,39)(21,51,53,62,54,36,52,46)(27,40,57,55,59,64,58,60)(48,63,67,66,68,65,56,61);(1,2,4,3)(5,7)(6,13,18,38)(8,26,30,49)(9,21)(10,22)(11,23)(12,24)(14,39)(15,36,44,51)(16,19,17,37)(20,29,50,28)(25,47)(27,56,59,67)(31,46,43,62)(32,52)(33,53)(34,54)(35,41)(40,63)(42,45)(48,58,68,57)(55,66)(60,61)(64,65);(5,8)(6,14,18,42)(7,20,24,50)(9,27)(10,28)(11,29)(12,30)(13,35)(15,40,44,64)(16,41,17,25)(19,45)(21,48,54,68)(22,49,23,26)(31,60,43,55)(32,57)(33,58)(34,59)(36,61)(37,39)(38,47)(46,65)(51,66)(52,67,53,56)(62,63);(1,3,4,2)(5,9,11,33,12,34,10,32)(6,15,17,43,18,44,16,31)(7,21,23,53,24,54,22,52)(8,27,29,58,30,59,28,57)(13,36,37,62,38,51,19,46)(14,40,25,55,42,64,41,60)(20,48,26,56,50,68,49,67)(35,61,39,63,47,66,45,65)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=fg,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276383);