Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360200 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2311 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13925 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
907 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)(4,19,23,48)(5,24)(6,28)(7,12)(8,13)(9,17)(10,22,36,21)(11,25)(15,30,42,34)(16,43)(18,32)(20,49,52,35)(26,37)(27,54)(29,33)(31,45)(38,41,46,53)(39,44)(40,59,61,62)(47,51,63,50)(55,57)(56,64,58,60);(1,3)(2,9,13,33)(4,15)(5,16)(6,17)(7,29)(8,18)(10,34,36,30)(11,31,37,43)(12,14,28,32)(19,46,48,38)(20,40)(21,41)(22,53)(23,42)(24,39,54,57)(25,44)(26,55)(27,45)(35,58,49,56)(47,62,63,59)(50,60)(51,64)(52,61);(1,4)(2,10,13,36)(3,15)(5,20)(6,21)(7,22)(8,23)(9,30,33,34)(11,35,37,49)(12,19,28,48)(14,38,32,46)(16,40)(17,41)(18,42)(24,47,54,63)(25,50)(26,51)(27,52)(29,53)(31,56,43,58)(39,59,57,62)(44,60)(45,61)(55,64);(1,5,6,25,8,27,7,26)(2,11,12,24,13,37,28,54)(3,16,17,44,18,45,29,55)(4,20,21,50,23,52,22,51)(9,31,14,39,33,43,32,57)(10,35,19,47,36,49,48,63)(15,40,41,60,42,61,53,64)(30,56,38,59,34,58,46,62)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,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,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 270109);