Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #27987 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2504 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13881 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1171 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)(9,23,34,41)(10,26)(11,28)(15,40,21,29)(16,45)(18,31)(19,37)(22,32,53,57)(27,35)(30,48)(36,55)(38,44,49,43)(39,46)(42,62,51,59)(47,58)(50,61,54,52)(56,63,60,64);(1,3,6,17,7,19,28,18)(2,8,11,14,13,31,12,37)(4,21,23,44,24,15,41,43)(5,16,26,46,27,48,55,47)(9,29,33,38,34,40,20,49)(10,30,25,39,36,45,35,58)(22,51,52,64,53,42,61,63)(32,56,54,59,57,60,50,62);(1,4,7,24)(2,9)(3,15)(5,22,27,53)(6,23,28,41)(8,29,31,40)(10,32)(11,33)(12,20)(13,34)(14,38,37,49)(16,42)(17,43)(18,44)(19,21)(25,54)(26,52,55,61)(30,56,45,60)(35,50)(36,57)(39,59,58,62)(46,63)(47,64)(48,51);(1,5,6,26,7,27,28,55)(2,10,11,25,13,36,12,35)(3,16,17,46,19,48,18,47)(4,22,23,52,24,53,41,61)(8,30,14,39,31,45,37,58)(9,32,33,54,34,57,20,50)(15,42,43,63,21,51,44,64)(29,56,38,59,40,60,49,62)Polycyclic presentation:
2^9|a^2=h,b^2=e,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, 268020);