Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #363904 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2297 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13959 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1177 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)(3,13,17,36)(4,18,21,9)(5,22,24,51)(6,11)(7,12)(8,16,28,43)(10,23,33,52)(14,34)(15,40)(19,46)(20,45,49,31)(25,32)(26,53)(27,42)(29,38)(30,47)(35,58)(37,55,61,54)(39,56)(41,59)(44,62)(48,50)(57,60,63,64);(1,3)(2,8,12,28)(4,14)(5,15)(6,16)(7,17)(9,29,18,26)(10,27,33,40)(11,13,32,36)(19,37)(20,38)(21,39)(22,35,51,59)(23,41)(24,42)(25,43)(30,55,44,54)(31,56,45,34)(46,57,50,63)(47,60)(48,61)(49,53)(52,58)(62,64);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,26)(10,30)(11,31)(12,18)(13,34)(15,37)(16,38)(17,39)(22,50)(23,47)(24,48)(25,49)(27,54)(28,29)(32,45)(33,44)(35,57)(36,56)(40,55)(41,60)(42,61)(43,53)(46,51)(52,62)(58,64)(59,63);(1,5,6,23,7,24,25,52)(2,10,32,51,12,33,11,22)(3,15,16,41,17,42,43,58)(4,19,20,47,21,48,49,62)(8,27,36,59,28,40,13,35)(9,30,45,46,18,44,31,50)(14,37,38,60,39,61,53,64)(26,54,56,63,29,55,34,57)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=eg,e^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^b=hi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271296);