Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #392203 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2280 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14680 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3337 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)(4,20,23,34)(5,24,27,11)(6,28)(7,12)(8,13)(9,17)(10,22,35,29)(15,30,42,58)(16,43,46,38)(18,32)(19,39)(21,50)(25,36,26,55)(31,44,53,45)(33,51)(37,41,48,47)(40,60)(49,52)(54,56)(57,62)(59,61)(63,64);(1,3,6,17,8,19,7,18)(2,9,12,14,13,32,28,39)(4,15,22,41,23,42,29,47)(5,16,25,44,27,46,26,45)(10,30,34,37,35,58,20,48)(11,31,36,38,24,53,55,43)(21,40,51,61,52,62,56,64)(33,57,50,59,54,60,49,63);(1,4)(2,10)(3,15)(5,21)(6,22)(7,29)(8,23)(9,30)(11,33)(12,34)(13,35)(14,37)(16,40)(17,41)(18,47)(19,42)(20,28)(24,54)(25,51)(26,56)(27,52)(31,57)(32,58)(36,50)(38,59)(39,48)(43,63)(44,61)(45,64)(46,62)(49,55)(53,60);(1,5)(2,11)(3,16)(4,21)(6,25)(7,26)(8,27)(9,31)(10,33)(12,36)(13,24)(14,38)(15,40)(17,44)(18,45)(19,46)(20,49)(22,51)(23,52)(28,55)(29,56)(30,57)(32,53)(34,50)(35,54)(37,59)(39,43)(41,61)(42,62)(47,64)(48,63)(58,60)Polycyclic presentation:
2^9|a^2=h,b^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi,f^c=fi,h^c=hi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299399);