Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437798 |
| 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×#3586 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4460 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)(5,6,10,16,12,18,11,17)(7,19,22,37,24,36,23,13)(8,14,28,41,27,40,32,45)(9,15,30,43,31,44,33,46)(20,38,25,52,48,59,50,34)(21,42,29,57,49,60,51,35)(26,39,55,63,54,62,56,64)(47,61,53,67,65,68,66,58);(5,7,12,24)(6,13,18,37)(8,25,27,50)(9,29,31,51)(10,23,11,22)(14,38,40,59)(15,42,44,60)(16,36,17,19)(20,32,48,28)(21,33,49,30)(26,47,54,65)(34,45,52,41)(35,46,57,43)(39,58,62,67)(53,55,66,56)(61,63,68,64);(2,4)(5,8)(6,14)(7,20)(9,26)(10,28)(11,32)(12,27)(13,34)(15,39)(16,41)(17,45)(18,40)(19,38)(21,47)(22,25)(23,50)(24,48)(29,53)(30,55)(31,54)(33,56)(35,58)(36,59)(37,52)(42,61)(43,63)(44,62)(46,64)(49,65)(51,66)(57,67)(60,68);(5,9)(6,15)(7,21)(8,26)(10,30)(11,33)(12,31)(13,35)(14,39)(16,43)(17,46)(18,44)(19,42)(20,47)(22,29)(23,51)(24,49)(25,53)(27,54)(28,55)(32,56)(34,58)(36,60)(37,57)(38,61)(40,62)(41,63)(45,64)(48,65)(50,66)(52,67)(59,68);(8,27)(14,40)(20,48)(25,50)(26,54)(28,32)(34,52)(38,59)(39,62)(41,45)(47,65)(53,66)(55,56)(58,67)(61,68)(63,64)Polycyclic presentation:
2^9|a^2=f,b^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344996);