Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455561 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,10,16,12,17,11,31)(7,13,20,36,21,38,32,37)(8,22,24,41,25,40,30,14)(9,26,29,15,28,44,33,43)(18,45,47,61,49,64,48,34)(19,50,53,63,52,58,54,35)(23,42,56,59,27,39,57,55)(46,60,66,65,51,62,67,68);(1,2,3,4)(5,7)(6,13)(8,18)(9,19)(10,20)(11,32)(12,21)(14,34)(15,35)(16,36)(17,38)(22,45)(23,46)(24,47)(25,49)(26,58)(27,51)(28,52)(29,53)(30,48)(31,37)(33,54)(39,60)(40,64)(41,61)(42,62)(43,63)(44,50)(55,65)(56,66)(57,67)(59,68);(2,4)(5,8,12,25)(6,14,17,41)(7,18,21,49)(9,27,28,23)(10,30,11,24)(13,34,38,61)(15,42,43,39)(16,40,31,22)(19,51,52,46)(20,48,32,47)(26,59,44,55)(29,56,33,57)(35,62,63,60)(36,64,37,45)(50,65,58,68)(53,66,54,67);(5,9,10,29,12,28,11,33)(6,15,16,44,17,43,31,26)(7,19,20,53,21,52,32,54)(8,23,24,56,25,27,30,57)(13,35,36,50,38,63,37,58)(14,39,22,55,41,42,40,59)(18,46,47,66,49,51,48,67)(34,60,45,65,61,62,64,68)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,d^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^b=hi,d^c=de,f^c=fi,e^d=ei,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 373029);