Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #27987 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2504 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14002 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
965 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,14)(4,20,24,33)(5,25)(6,12)(7,13)(8,17)(9,23,34,53)(10,26)(11,28)(15,40,21,29)(16,44)(18,31)(19,37)(22,32,52,57)(27,35)(30,47)(36,55)(38,43,48,42)(39,45)(41,61,50,59)(46,58)(49,64,54,51)(56,62,60,63);(1,3)(2,8,13,31)(4,21)(5,16)(6,17)(7,19)(9,29,34,40)(10,30,36,44)(11,14,12,37)(15,24)(18,28)(20,48,33,38)(22,50)(23,43)(25,39,35,58)(26,45)(27,47)(32,56,57,60)(41,52)(42,53)(46,55)(49,61,54,59)(51,63)(62,64);(1,4,7,24)(2,9)(3,15)(5,22,27,52)(6,23,28,53)(8,29,31,40)(10,32)(11,33)(12,20)(13,34)(14,38,37,48)(16,41)(17,42)(18,43)(19,21)(25,54)(26,51,55,64)(30,56,44,60)(35,49)(36,57)(39,59,58,61)(45,62)(46,63)(47,50);(1,5,6,26,7,27,28,55)(2,10,11,25,13,36,12,35)(3,16,17,45,19,47,18,46)(4,22,23,51,24,52,53,64)(8,30,14,39,31,44,37,58)(9,32,33,54,34,57,20,49)(15,41,42,62,21,50,43,63)(29,56,38,59,40,60,48,61)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=ef,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274577);