Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29902 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,11,18,12,16,10,17)(7,19,23,36,24,13,22,37)(8,25,29,40,30,14,28,41)(9,15,33,45,31,43,32,44)(20,34,49,54,50,46,26,38)(21,47,53,61,51,35,52,62)(27,42,58,64,56,39,57,65)(48,59,68,66,67,60,55,63);(1,2)(3,4)(5,7,12,24)(6,13,16,19)(8,26,30,49)(9,21,31,51)(10,23,11,22)(14,38,25,54)(15,35,43,47)(17,37,18,36)(20,29,50,28)(27,55,56,68)(32,53,33,52)(34,41,46,40)(39,63,42,66)(44,62,45,61)(48,58,67,57)(59,65,60,64);(2,4)(5,8,12,30)(6,14,16,25)(7,20,24,50)(9,27,31,56)(10,29,11,28)(13,34,19,46)(15,42,43,39)(17,41,18,40)(21,48,51,67)(22,49,23,26)(32,58,33,57)(35,60,47,59)(36,54,37,38)(44,64,45,65)(52,68,53,55)(61,63,62,66);(5,9)(6,15)(7,21)(8,27)(10,32)(11,33)(12,31)(13,35)(14,39)(16,43)(17,44)(18,45)(19,47)(20,48)(22,52)(23,53)(24,51)(25,42)(26,55)(28,57)(29,58)(30,56)(34,59)(36,61)(37,62)(38,63)(40,64)(41,65)(46,60)(49,68)(50,67)(54,66)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,g^b=gi,h^b=hi,d^c=de,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, 287031);