Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360200 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2311 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13872 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1553 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)(4,19,22,36)(5,23)(6,27)(7,12)(8,13)(9,17)(10,21,37,29)(11,24)(15,34,44,30)(16,45)(18,32)(20,35,50,59)(25,38)(26,52)(28,33)(31,47)(39,43,57,42)(40,46)(41,61,64,60)(48,54,51,49)(53,56)(55,62,58,63);(1,3,6,17,8,18,7,28)(2,9,12,14,13,33,27,32)(4,15,21,42,22,44,29,43)(5,16,24,46,26,47,25,53)(10,34,36,57,37,30,19,39)(11,31,23,40,38,45,52,56)(20,41,49,62,50,64,54,63)(35,58,51,61,59,55,48,60);(1,4)(2,10)(3,15)(5,20)(6,21)(7,29)(8,22)(9,30)(11,35)(12,36)(13,37)(14,39)(16,41)(17,42)(18,44)(19,27)(23,51)(24,49)(25,54)(26,50)(28,43)(31,55)(32,57)(33,34)(38,59)(40,60)(45,58)(46,62)(47,64)(48,52)(53,63)(56,61);(1,5,6,24,8,26,7,25)(2,11,12,23,13,38,27,52)(3,16,17,46,18,47,28,53)(4,20,21,49,22,50,29,54)(9,31,14,40,33,45,32,56)(10,35,36,51,37,59,19,48)(15,41,42,62,44,64,43,63)(30,55,39,60,34,58,57,61)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,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, 267410);