Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418851 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2679 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15961 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1051 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,26,6,12,18,11)(3,8,28,31,15,32,16,44)(4,17,27,35,21,38,20,9)(5,22,25,37,24,10,36,53)(13,39,41,29,42,46,45,33)(14,30,55,58,23,51,43,62)(19,48,56,34,50,47,49,52)(40,60,61,57,54,64,63,59);(1,3)(2,8)(4,13)(5,23)(6,15)(7,16)(9,33)(10,30)(11,31)(12,32)(14,24)(17,46)(18,28)(19,40)(20,41)(21,42)(22,51)(25,55)(26,44)(27,45)(29,35)(34,57)(36,43)(37,58)(38,39)(47,60)(48,64)(49,61)(50,54)(52,59)(53,62)(56,63);(1,4,18,20,6,21,7,27)(2,9,11,38,12,35,26,17)(3,13,28,41,15,42,16,45)(5,19,36,49,24,50,25,56)(8,29,31,46,32,33,44,39)(10,34,37,48,22,52,53,47)(14,40,55,61,23,54,43,63)(30,57,58,64,51,59,62,60);(1,5)(2,10)(3,14)(4,19)(6,24)(7,25)(8,30)(9,34)(11,37)(12,22)(13,40)(15,23)(16,43)(17,47)(18,36)(20,49)(21,50)(26,53)(27,56)(28,55)(29,57)(31,58)(32,51)(33,59)(35,52)(38,48)(39,60)(41,61)(42,54)(44,62)(45,63)(46,64)Polycyclic presentation:
2^9|a^2=efh,c^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,h^b=hi,e^c=ei,f^c=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 357420);