Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395586 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2411 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14923 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
837 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,11,7,12,25,33)(3,13,34,8,17,37,16,28)(4,9,20,30,21,31,48,57)(5,22,49,32,24,10,23,52)(14,38,45,54,18,35,40,26)(15,41,42,27,43,36,58,50)(19,29,55,64,47,51,46,56)(39,60,61,53,44,59,62,63);(1,3)(2,8,12,28)(4,18,21,14)(5,15)(6,16)(7,17)(9,26)(10,27,22,50)(11,13,33,37)(19,44,47,39)(20,45,48,40)(23,42)(24,43)(25,34)(29,53)(30,35)(31,54)(32,36,52,41)(38,57)(46,62,55,61)(49,58)(51,63)(56,59)(60,64);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,26)(10,29)(11,30)(12,31)(13,35)(15,39)(16,40)(17,18)(22,51)(23,46)(24,47)(25,48)(27,53)(28,54)(32,56)(33,57)(34,45)(36,59)(37,38)(41,60)(42,61)(43,44)(49,55)(50,63)(52,64)(58,62);(1,5,6,23,7,24,25,49)(2,10,11,32,12,22,33,52)(3,15,16,42,17,43,34,58)(4,19,20,46,21,47,48,55)(8,27,13,36,28,50,37,41)(9,29,30,56,31,51,57,64)(14,39,40,61,18,44,45,62)(26,53,35,59,54,63,38,60)Polycyclic presentation:
2^9|a^2=ef,b^2=h,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 313257);