Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455564 |
| 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 | |
1045 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)(7,12,20,35)(8,21,24,40)(9,25,27,14)(10,29,31,15)(13,30,37,23)(17,41,44,62)(18,33,46,54)(19,48,50,34)(22,52,26,51)(28,39,57,55)(32,49,59,43)(36,53,38,56)(42,63,45,65)(47,68,67,61)(58,66,60,64);(1,2,3,4)(5,7)(6,12)(8,17)(9,18)(10,19)(11,20)(13,32)(14,33)(15,34)(16,35)(21,41)(22,42)(23,43)(24,44)(25,54)(26,45)(27,46)(28,47)(29,48)(30,49)(31,50)(36,58)(37,59)(38,60)(39,61)(40,62)(51,63)(52,65)(53,64)(55,68)(56,66)(57,67);(2,4)(5,8,11,24)(6,13,16,37)(7,17,20,44)(9,26,27,22)(10,30,31,23)(12,32,35,59)(14,38,25,36)(15,40,29,21)(18,45,46,42)(19,49,50,43)(28,53,57,56)(33,60,54,58)(34,62,48,41)(39,51,55,52)(47,64,67,66)(61,63,68,65);(5,9,10,28,11,27,31,57)(6,14,15,39,16,25,29,55)(7,18,19,47,20,46,50,67)(8,22,23,53,24,26,30,56)(12,33,34,61,35,54,48,68)(13,36,21,51,37,38,40,52)(17,42,43,64,44,45,49,66)(32,58,41,63,59,60,62,65)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=i,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,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, 372821);