Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #400051 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2535 |
| 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 | |
1270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,4)(5,6)(7,18,23,13)(8,24)(9,29)(10,16)(11,17)(12,34)(14,27)(15,32)(19,43,47,35)(20,48,51,59)(21,45,22,37)(25,53,52,38)(26,56)(28,41)(30,61)(31,42)(33,40)(36,50,44,49)(39,62)(46,64,66,68)(54,58)(55,65,67,63)(57,60);(1,2,3,4)(5,7,12,22,10,23,11,21)(6,13,34,45,16,18,17,37)(8,25,33,19,27,52,28,47)(9,20,32,50,30,51,31,49)(14,38,41,35,24,53,40,43)(15,36,61,48,42,44,29,59)(26,55,62,46,57,67,58,66)(39,64,60,63,54,68,56,65);(5,8)(6,14)(7,19)(9,26)(10,27)(11,33)(12,28)(13,35)(15,39)(16,24)(17,41)(18,43)(20,46)(21,52)(22,25)(23,47)(29,60)(30,57)(31,62)(32,58)(34,40)(36,63)(37,53)(38,45)(42,54)(44,65)(48,64)(49,67)(50,55)(51,66)(56,61)(59,68);(5,9,11,31,10,30,12,32)(6,15,34,61,16,42,17,29)(7,20,21,49,23,51,22,50)(8,26,33,62,27,57,28,58)(13,36,45,48,18,44,37,59)(14,39,40,56,24,54,41,60)(19,46,52,67,47,66,25,55)(35,63,38,64,43,65,53,68)Polycyclic presentation:
2^9|a^2=fi,b^2=eh,d^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 323403);