Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #365053 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2363 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13970 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1615 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)(4,18,22,9)(5,23)(6,26)(7,12)(8,16)(10,24)(11,27)(14,38,19,35)(15,41)(17,37)(20,50,52,53)(21,33,47,46)(25,54)(28,40,31,49)(29,43)(30,44)(32,51,45,63)(34,55)(36,42)(39,56,48,57)(58,61,60,64)(59,62);(1,3,6,16,7,17,27,44)(2,8,11,13,12,30,26,37)(4,19,47,40,22,14,21,49)(5,15,24,42,25,43,55,62)(9,31,46,35,18,28,33,38)(10,29,23,36,34,41,54,59)(20,48,63,61,52,39,51,64)(32,57,50,58,45,56,53,60);(1,4)(2,9)(3,14)(5,20)(6,21)(7,22)(8,28)(10,32)(11,33)(12,18)(13,35)(15,39)(16,40)(17,19)(23,53)(24,51)(25,52)(26,46)(27,47)(29,56)(30,31)(34,45)(36,58)(37,38)(41,57)(42,61)(43,48)(44,49)(50,54)(55,63)(59,60)(62,64);(1,5,6,24,7,25,27,55)(2,10,11,23,12,34,26,54)(3,15,16,42,17,43,44,62)(4,20,21,51,22,52,47,63)(8,29,13,36,30,41,37,59)(9,32,33,53,18,45,46,50)(14,39,40,61,19,48,49,64)(28,56,35,58,31,57,38,60)Polycyclic presentation:
2^9|a^2=h,b^2=eh,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,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272249);