Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6629034 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(1,2)(3,4)(5,6,12,18)(7,19,24,35)(8,25,28,42)(9,15,29,39)(10,17,11,16)(13,23,36,22)(14,32,38,27)(20,45,48,33)(21,44,49,59)(26,53,54,62)(30,41,31,40)(34,51,58,50)(37,56,61,55)(43,52,60,47)(46,64,65,57)(63,67,68,66);(5,7,11,23,12,24,10,22)(6,13,17,35,18,36,16,19)(8,20,32,52,28,48,27,47)(9,21,31,51,29,49,30,50)(14,33,42,60,38,45,25,43)(15,34,41,59,39,58,40,44)(26,46,56,67,54,65,55,66)(37,57,62,68,61,64,53,63);(5,8)(6,14)(7,20)(9,26)(10,27)(11,32)(12,28)(13,33)(15,37)(16,25)(17,42)(18,38)(19,43)(21,46)(22,47)(23,52)(24,48)(29,54)(30,55)(31,56)(34,57)(35,60)(36,45)(39,61)(40,53)(41,62)(44,63)(49,65)(50,66)(51,67)(58,64)(59,68);(2,4)(5,9)(6,15)(7,21)(8,26)(10,30)(11,31)(12,29)(13,34)(14,37)(16,40)(17,41)(18,39)(19,44)(20,46)(22,50)(23,51)(24,49)(25,53)(27,55)(28,54)(32,56)(33,57)(35,59)(36,58)(38,61)(42,62)(43,63)(45,64)(47,66)(48,65)(52,67)(60,68);(9,29)(15,39)(21,49)(26,54)(30,31)(34,58)(37,61)(40,41)(44,59)(46,65)(50,51)(53,62)(55,56)(57,64)(63,68)(66,67)Polycyclic presentation:
2^9|a^2=i,b^2=fi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,g^c=gi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536232);