Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395163 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1692 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,11,16)(7,17,22,40)(8,13,25,36)(9,26,28,14)(10,29,18,15)(12,30,33,21)(19,38,45,61)(20,46,48,39)(23,34,50,53)(24,51,41,35)(27,37,42,54)(31,55,58,44)(32,47,52,56)(43,60,64,65)(49,66,62,59)(57,68,67,63);(1,2)(3,4)(5,7,18,30,11,22,10,21)(6,12,29,40,16,33,15,17)(8,19,41,55,25,45,24,44)(9,20,42,56,28,48,27,47)(13,31,51,61,36,58,35,38)(14,32,54,46,26,52,37,39)(23,43,62,68,50,64,49,63)(34,57,66,65,53,67,59,60);(2,4)(5,8)(6,13)(7,19)(9,23)(10,24)(11,25)(12,31)(14,34)(15,35)(16,36)(17,38)(18,41)(20,43)(21,44)(22,45)(26,53)(27,49)(28,50)(29,51)(30,55)(32,57)(33,58)(37,59)(39,60)(40,61)(42,62)(46,65)(47,63)(48,64)(52,67)(54,66)(56,68);(5,9,10,27,11,28,18,42)(6,14,15,37,16,26,29,54)(7,20,21,47,22,48,30,56)(8,23,24,49,25,50,41,62)(12,32,17,39,33,52,40,46)(13,34,35,59,36,53,51,66)(19,43,44,63,45,64,55,68)(31,57,38,60,58,67,61,65)Polycyclic presentation:
2^9|a^2=i,b^2=ei,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302376);