Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24808 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2416 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14046 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
827 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)(4,19,23,9)(5,24)(6,27)(7,12)(8,16)(10,25)(11,28)(14,37,41,33)(15,42)(17,31)(18,32)(20,48,40,29)(21,50,52,54)(22,35,53,47)(26,55)(30,45)(34,51,46,64)(36,56)(38,43)(39,61,63,57)(44,58)(49,59,62,60);(1,3)(2,8,12,31)(4,20)(5,15)(6,16)(7,18)(9,33,19,37)(10,30,36,42)(11,13,27,32)(14,22)(17,28)(21,49)(23,40)(24,38,55,58)(25,43)(26,45)(29,47,48,35)(34,59,46,60)(39,51)(41,53)(44,56)(50,61,54,57)(52,62)(63,64);(1,4,6,22,7,23,28,53)(2,9,11,35,12,19,27,47)(3,14,17,20,18,41,16,40)(5,21,25,51,26,52,56,64)(8,29,32,33,31,48,13,37)(10,34,24,54,36,46,55,50)(15,39,44,49,45,63,43,62)(30,57,58,59,42,61,38,60);(1,5,6,25,7,26,28,56)(2,10,11,24,12,36,27,55)(3,15,16,43,18,45,17,44)(4,21,22,51,23,52,53,64)(8,30,13,38,31,42,32,58)(9,34,35,54,19,46,47,50)(14,39,40,62,41,63,20,49)(29,57,37,60,48,61,33,59)Polycyclic presentation:
2^9|a^2=h,b^2=efgh,c^2=g,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 277486);