Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1553 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)(7,18,23,44)(8,24,29,53)(9,30)(10,16)(11,35)(12,17)(13,21,38,22)(14,27,41,28)(15,34)(19,45)(20,49,51,42)(25,39)(26,55,58,52)(31,37,50,63)(32,60)(33,43)(36,48)(40,57,59,56)(46,62)(47,65)(54,68)(61,64)(66,67);(1,2)(3,4)(5,7)(6,13)(8,25)(9,31)(10,22)(11,21)(12,23)(14,39)(15,42)(16,44)(17,38)(18,35)(19,28)(20,32)(24,36)(26,46)(27,48)(29,47)(30,49)(33,51)(34,50)(37,43)(40,66)(41,65)(45,53)(52,64)(54,56)(55,68)(57,61)(58,67)(59,62)(60,63);(5,8,11,28,12,29,10,27)(6,14,16,24,17,41,35,53)(7,19,22,25,23,48,21,47)(9,26,33,57,34,58,32,56)(13,36,18,39,38,45,44,65)(15,40,43,52,30,59,60,55)(20,46,50,54,51,67,31,61)(37,62,42,64,63,66,49,68);(1,3,4,2)(5,9,11,33,12,34,10,32)(6,15,35,60,17,30,16,43)(7,20,22,50,23,51,21,31)(8,26,28,57,29,58,27,56)(13,37,44,49,38,63,18,42)(14,40,53,55,41,59,24,52)(19,46,25,54,48,67,47,61)(36,62,65,68,45,66,39,64)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293286);