Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384436 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
921 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)(5,23,26,37)(6,27)(7,12)(8,16)(10,25,38,44)(11,29)(14,41,19,40)(15,31,45,55)(17,39)(20,50)(21,35,54,47)(24,57,59,36)(28,32)(30,43,33,49)(34,52)(42,61)(46,64)(48,60)(51,62)(53,56)(58,63);(1,3,7,17)(2,8,12,32)(4,19,22,14)(5,24,26,59)(6,28,29,16)(9,33,18,30)(10,36,38,57)(11,39,27,13)(15,44,45,25)(20,51,53,58)(21,43,54,49)(23,55,37,31)(34,62,46,63)(35,40,47,41)(42,52,48,64)(50,61,56,60);(1,4,6,21,7,22,29,54)(2,9,27,47,12,18,11,35)(3,14,16,43,17,19,28,49)(5,20,25,52,26,53,44,64)(8,30,39,41,32,33,13,40)(10,34,23,56,38,46,37,50)(15,42,24,58,45,48,59,51)(31,60,57,62,55,61,36,63);(1,5,6,25,7,26,29,44)(2,10,11,37,12,38,27,23)(3,15,16,24,17,45,28,59)(4,20,21,52,22,53,54,64)(8,31,13,36,32,55,39,57)(9,34,35,50,18,46,47,56)(14,42,43,58,19,48,49,51)(30,60,40,63,33,61,41,62)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=ef,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^c=gi,h^c=hi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291633);