Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360368 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2344 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13921 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
679 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)(3,14,19,9)(4,20)(5,25,28,11)(6,29)(7,13)(8,12)(10,23)(15,34,43,57)(16,32)(17,33,18,41)(21,39,47,31)(22,50,52,48)(24,37)(26,38,27,54)(30,36)(35,51,53,55)(40,46)(42,62)(44,61)(45,58)(49,56)(59,63)(60,64);(1,3)(2,9)(4,21,24,47)(5,16)(6,17)(7,18)(8,19)(10,34,36,57)(11,32)(12,14)(13,33)(15,23,43,30)(20,31,37,39)(22,49,52,64)(25,40)(26,44)(27,45)(28,46)(29,41)(35,59,53,62)(38,58)(42,51,63,55)(48,56,50,60)(54,61);(1,4)(2,10,12,36)(3,15)(5,22)(6,23)(7,30)(8,24)(9,31,14,39)(11,35,25,53)(13,20,29,37)(16,42)(17,21)(18,47)(19,43)(26,51)(27,55)(28,52)(32,56,40,60)(33,57,41,34)(38,48,54,50)(44,49)(45,64)(46,63)(58,62,61,59);(1,5,6,26,8,28,7,27)(2,11,29,54,12,25,13,38)(3,16,17,44,19,46,18,45)(4,22,23,51,24,52,30,55)(9,32,41,61,14,40,33,58)(10,35,37,50,36,53,20,48)(15,42,21,49,43,63,47,64)(31,56,34,59,39,60,57,62)Polycyclic presentation:
2^9|a^2=h,b^2=efg,c^2=e,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,h^b=hi,g^c=gi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269891);