Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360200 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2311 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13872 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1541 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,14,6,13)(3,15,19,32,20,39,18,9)(4,21,23,10,25,36,24,35)(5,26,28,38,29,37,27,11)(16,30,43,50,44,33,49,40)(17,45,47,59,48,31,46,41)(22,52,53,51,55,34,54,58)(42,60,61,56,62,64,63,57);(1,3,6,18,8,20,7,19)(2,9,12,15,14,32,13,39)(4,16,23,43,25,44,24,49)(5,17,27,46,29,48,28,47)(10,33,21,50,35,30,36,40)(11,31,26,41,38,45,37,59)(22,42,53,61,55,62,54,63)(34,57,51,64,52,56,58,60);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,25)(9,30)(11,34)(12,21)(13,36)(14,35)(15,40)(17,42)(18,43)(19,49)(20,44)(26,51)(27,53)(28,54)(29,55)(31,56)(32,33)(37,58)(38,52)(39,50)(41,60)(45,57)(46,61)(47,63)(48,62)(59,64);(1,5,6,27,8,29,7,28)(2,11,12,26,14,38,13,37)(3,17,18,46,20,48,19,47)(4,22,23,53,25,55,24,54)(9,31,15,41,32,45,39,59)(10,34,21,51,35,52,36,58)(16,42,43,61,44,62,49,63)(30,56,40,60,33,57,50,64)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267411);