Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21782 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2308 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13794 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2345 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)(3,14,18,9)(4,19,22,47)(5,24,26,11)(6,13)(7,29)(8,12)(10,21,36,23)(15,39)(16,31)(17,33,28,32)(20,48)(25,54,27,37)(30,41)(34,52)(35,49)(38,44)(40,61,63,60)(42,45)(43,57)(46,51)(50,53)(55,58)(56,62,59,64);(1,3)(2,9)(4,15)(5,16)(6,28)(7,17)(8,18)(10,34)(11,31)(12,14)(13,33)(19,45)(20,40)(21,52)(22,42)(23,41)(24,38)(25,55)(26,44)(27,43)(29,32)(30,36)(35,59)(37,58)(39,47)(46,61)(48,60)(49,64)(50,62)(51,63)(53,56)(54,57);(1,4,7,23,8,22,6,21)(2,10,13,19,12,36,29,47)(3,15,17,41,18,42,28,52)(5,20,27,50,26,51,25,49)(9,30,33,39,14,34,32,45)(11,35,37,46,24,53,54,48)(16,40,43,62,44,63,55,64)(31,56,58,60,38,59,57,61);(1,5)(2,11)(3,16)(4,20)(6,25)(7,27)(8,26)(9,31)(10,35)(12,24)(13,37)(14,38)(15,40)(17,43)(18,44)(19,46)(21,49)(22,51)(23,50)(28,55)(29,54)(30,56)(32,57)(33,58)(34,59)(36,53)(39,60)(41,62)(42,63)(45,61)(47,48)(52,64)Polycyclic presentation:
2^9|a^2=g,c^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,h^a=hi,c^b=ce,f^b=fi,g^b=gi,g^c=gi,g^d=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263121);