Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405373 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2853 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14912 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1222 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)(5,11)(6,28)(7,12)(8,13)(9,17)(10,23)(15,34)(16,40)(18,32)(20,47)(21,35)(22,36)(24,46)(25,53)(26,38)(27,37)(29,33)(30,49)(31,43)(39,42)(41,60)(44,57)(45,52)(48,55)(50,62)(51,59)(54,56)(58,61)(63,64);(1,3)(2,9)(4,20,23,49)(5,16)(6,29)(7,17)(8,18)(10,34,19,39)(11,31)(12,32)(13,33)(14,28)(15,24,42,22)(21,48,52,64)(25,54)(26,44)(27,43)(30,36,47,46)(35,58,45,60)(37,57)(38,56)(40,53)(41,51,61,50)(55,59,63,62);(1,4,6,22,8,23,7,24)(2,10,12,36,13,19,28,46)(3,15,17,20,18,42,29,49)(5,21,25,50,26,52,27,51)(9,30,14,39,33,47,32,34)(11,35,37,59,38,45,53,62)(16,41,43,48,44,61,54,64)(31,55,40,60,56,63,57,58);(1,5)(2,11)(3,16)(4,21)(6,25)(7,27)(8,26)(9,31)(10,35)(12,37)(13,38)(14,40)(15,41)(17,43)(18,44)(19,45)(20,48)(22,50)(23,52)(24,51)(28,53)(29,54)(30,55)(32,57)(33,56)(34,58)(36,59)(39,60)(42,61)(46,62)(47,63)(49,64)Polycyclic presentation:
2^9|b^2=h,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^c=fi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312569);