Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384777 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2337 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14477 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
585 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)(3,14,19,41)(4,20)(5,26,29,39)(6,12)(7,13)(8,31)(9,17,34,18)(10,24)(11,30,40,28)(15,35,44,48)(16,45)(21,32,43,60)(22,52,54,56)(23,50)(25,37)(27,57)(33,47)(36,53,49,55)(38,46)(42,62)(51,61)(58,64)(59,63);(1,3,8,17,7,19,6,18)(2,9,31,14,13,34,12,41)(4,21,25,15,24,43,23,44)(5,27,30,47,29,46,28,16)(10,35,50,60,20,48,37,32)(11,38,26,45,40,57,39,33)(22,42,55,58,54,63,53,51)(36,62,56,64,49,59,52,61);(1,4,6,23,7,24,8,25)(2,10,12,37,13,20,31,50)(3,15,17,43,19,44,18,21)(5,22,28,53,29,54,30,55)(9,32,14,35,34,60,41,48)(11,36,39,52,40,49,26,56)(16,42,27,58,47,63,46,51)(33,59,38,61,45,62,57,64);(1,5,8,30,7,29,6,28)(2,11,12,39,13,40,31,26)(3,16,18,46,19,47,17,27)(4,22,25,55,24,54,23,53)(9,33,14,38,34,45,41,57)(10,36,37,52,20,49,50,56)(15,42,21,51,44,63,43,58)(32,59,35,61,60,62,48,64)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,d^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294731);