Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360370 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13874 |
| commutator, prime radical | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1003 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,11,28,37)(3,13,17,38,16,8,26,31)(4,18,22,47,21,9,32,35)(5,23,24,39,25,10,27,36)(14,29,42,54,19,33,49,58)(15,30,44,55,43,40,45,56)(20,34,51,59,50,46,52,60)(41,57,61,64,48,53,62,63);(1,3)(2,8)(4,19,21,14)(5,15)(6,26)(7,16)(9,33,18,29)(10,30)(11,13)(12,38)(17,28)(20,48,50,41)(22,42,32,49)(23,40)(24,45)(25,43)(27,44)(31,37)(34,57,46,53)(35,54,47,58)(36,56)(39,55)(51,61,52,62)(59,63,60,64);(1,4)(2,9,11,18)(3,14)(5,20)(6,22)(7,21)(8,29,13,33)(10,34,23,46)(12,35,37,47)(15,41)(16,19)(17,42)(24,51)(25,50)(26,49)(27,52)(28,32)(30,53,40,57)(31,54,38,58)(36,59,39,60)(43,48)(44,61)(45,62)(55,63,56,64);(1,5,7,25)(2,10,11,23)(3,15,16,43)(4,20,21,50)(6,27,28,24)(8,30,13,40)(9,34,18,46)(12,39,37,36)(14,41,19,48)(17,45,26,44)(22,52,32,51)(29,53,33,57)(31,56,38,55)(35,60,47,59)(42,62,49,61)(54,64,58,63)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,h^b=hi,f^c=fi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267568);