Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458124 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2731 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16316 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
704 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)(5,22)(6,26)(7,12)(8,16)(9,21)(10,25)(11,28)(14,41)(15,45)(17,40)(19,51)(20,34)(23,35)(24,36)(27,37)(29,43)(30,59)(31,50)(32,49)(33,52)(38,44)(39,48)(42,60)(46,58)(47,53)(54,56)(55,57)(61,64)(62,63);(1,3,6,16,7,17,28,50)(2,8,11,13,12,31,26,40)(4,14,27,49,21,44,20,43)(5,15,24,47,25,48,57,59)(9,32,37,41,18,29,34,38)(10,30,36,39,22,53,55,45)(19,42,56,64,52,63,23,46)(33,61,54,60,51,58,35,62);(1,4,6,27,7,21,28,20)(2,9,11,37,12,18,26,34)(3,14,16,49,17,44,50,43)(5,23,24,19,25,56,57,52)(8,29,13,38,31,32,40,41)(10,35,36,33,22,54,55,51)(15,46,47,42,48,64,59,63)(30,60,39,58,53,62,45,61);(1,5,6,24,7,25,28,57)(2,10,11,36,12,22,26,55)(3,15,16,47,17,48,50,59)(4,19,20,23,21,52,27,56)(8,30,13,39,31,53,40,45)(9,33,34,35,18,51,37,54)(14,42,43,46,44,63,49,64)(29,58,41,60,32,61,38,62)Polycyclic presentation:
2^9|b^2=ef,c^2=ef,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,f^c=fi,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 365599);