Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #466 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1970 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(2,4)(5,6)(7,20)(8,27,32,46)(9,18)(10,17)(11,16)(12,36)(13,19)(14,22)(15,30,44,33)(21,50,53,64)(23,39)(24,48)(25,57)(26,49)(28,59,60,66)(29,58,63,42)(31,43,62,68)(34,45)(35,47)(37,51,65,61)(38,55)(40,56)(41,52,67,54);(1,2,3,4)(5,7,11,24,12,25,35,56,13,26,34,55,10,23,9,22)(6,14,18,39,17,38,45,49,19,40,47,57,36,48,16,20)(8,28,31,54,33,61,63,53,32,60,62,52,30,51,29,21)(15,41,43,59,27,50,58,65,44,67,68,66,46,64,42,37);(5,8)(6,15)(7,21)(9,31)(10,33)(11,29)(12,30)(13,32)(14,37)(16,43)(17,46)(18,42)(19,44)(20,41)(22,28)(23,54)(24,51)(25,52)(26,53)(27,36)(34,63)(35,62)(38,66)(39,64)(40,65)(45,68)(47,58)(48,59)(49,67)(50,57)(55,61)(56,60)Polycyclic presentation:
2^9|a^2=g,b^2=dh,d^2=hi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,d^c=dh,e^c=ei,f^c=fhi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55601);