Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25446 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2428 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14141 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1015 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,13)(4,9,21,33)(5,22)(6,25)(7,12)(8,16)(10,23)(11,26)(14,35,18,44)(15,40)(17,37)(19,47,49,64)(20,50,51,32)(24,52)(27,46,30,39)(28,42)(29,43)(31,56,57,48)(34,53)(36,41)(38,55,45,54)(58,63,62,60)(59,61);(1,3)(2,8,12,29)(4,18,21,14)(5,15)(6,16)(7,17)(9,30)(10,28,34,40)(11,13,25,37)(19,45,49,38)(20,46,51,39)(22,36,52,59)(23,41)(24,42)(26,43)(27,33)(31,55)(32,44)(35,50)(47,62)(48,63,56,60)(53,61)(54,57)(58,64);(1,4)(2,9,12,33)(3,14)(5,19)(6,20)(7,21)(8,27,29,30)(10,31,34,57)(11,32,25,50)(13,35,37,44)(15,38)(16,39)(17,18)(22,47,52,64)(23,48)(24,49)(26,51)(28,54,40,55)(36,58,59,62)(41,60)(42,45)(43,46)(53,56)(61,63);(1,5,6,23,7,24,26,53)(2,10,11,22,12,34,25,52)(3,15,16,41,17,42,43,61)(4,19,20,48,21,49,51,56)(8,28,13,36,29,40,37,59)(9,31,32,47,33,57,50,64)(14,38,39,60,18,45,46,63)(27,54,35,58,30,55,44,62)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fh,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281957);