Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #392203 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2280 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14685 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1265 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,13,7,14,6,12)(3,15,18,31,19,37,27,9)(4,10,21,33,22,34,28,38)(5,23,24,11,25,35,26,36)(16,39,44,29,43,54,42,55)(17,30,47,52,46,56,45,40)(20,48,51,58,50,57,49,32)(41,53,60,59,61,64,62,63);(1,3,8,27,7,19,6,18)(2,9,12,15,14,31,13,37)(4,16,28,44,22,43,21,42)(5,17,26,47,25,46,24,45)(10,29,33,39,34,55,38,54)(11,30,35,40,36,56,23,52)(20,41,51,62,50,61,49,60)(32,53,57,59,58,64,48,63);(1,4)(2,10)(3,16)(5,20)(6,21)(7,22)(8,28)(9,29)(11,32)(12,33)(13,38)(14,34)(15,39)(17,41)(18,42)(19,43)(23,48)(24,49)(25,50)(26,51)(27,44)(30,53)(31,55)(35,57)(36,58)(37,54)(40,59)(45,60)(46,61)(47,62)(52,63)(56,64);(1,5,6,24,7,25,8,26)(2,11,12,35,14,36,13,23)(3,17,18,45,19,46,27,47)(4,20,21,49,22,50,28,51)(9,30,15,40,31,56,37,52)(10,32,33,57,34,58,38,48)(16,41,42,60,43,61,44,62)(29,53,39,59,55,64,54,63)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,h^b=hi,h^c=hi,g^d=gi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299602);