Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #399126 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2504 |
| 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 | |
1372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,3)(5,6,12,18)(7,19,23,38)(8,25,29,14)(9,30,32,15)(10,17,11,16)(13,24,37,22)(20,39,47,53)(21,48,51,45)(26,35,52,61)(27,54,57,40)(28,41,34,44)(31,42,33,43)(36,49,58,50)(46,62,65,66)(55,60,67,68)(56,63,59,64);(1,2,4,3)(5,7,11,24,12,23,10,22)(6,13,17,38,18,37,16,19)(8,26,28,47,29,52,34,20)(9,21,33,50,32,51,31,49)(14,39,41,61,25,53,44,35)(15,36,43,48,30,58,42,45)(27,55,56,65,57,67,59,46)(40,62,63,68,54,66,64,60);(1,3,4,2)(5,8,10,28,12,29,11,34)(6,14,16,41,18,25,17,44)(7,20,22,26,23,47,24,52)(9,27,31,56,32,57,33,59)(13,35,19,39,37,61,38,53)(15,40,42,63,30,54,43,64)(21,46,49,55,51,65,50,67)(36,60,45,62,58,68,48,66);(5,9,12,32)(6,15,18,30)(7,21,23,51)(8,27,29,57)(10,31,11,33)(13,36,37,58)(14,40,25,54)(16,42,17,43)(19,45,38,48)(20,46,47,65)(22,49,24,50)(26,55,52,67)(28,56,34,59)(35,60,61,68)(39,62,53,66)(41,63,44,64)Polycyclic presentation:
2^9|a^2=i,b^2=eh,c^2=e,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi,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, 312683);