Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453815 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16125 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
680 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)(3,8,16,30)(4,9,19,33)(5,21)(6,11)(7,25)(10,24)(12,13)(14,28)(15,42,22,52)(17,36,26,31)(18,45,47,59)(20,49,38,34)(23,37)(27,35)(29,44,56,43)(32,51,57,48)(39,60)(40,55)(41,58)(46,63)(50,54)(53,61)(62,64);(1,3,13,17,6,16,7,26)(2,8,25,31,11,30,12,36)(4,14,38,41,19,40,20,50)(5,22,27,44,23,15,24,43)(9,28,49,54,33,55,34,58)(10,29,37,42,35,56,21,52)(18,46,51,62,47,39,48,61)(32,53,59,60,57,64,45,63);(1,4,6,19)(2,9,11,33)(3,14)(5,18,23,47)(7,20,13,38)(8,28)(10,32,35,57)(12,34,25,49)(15,39)(16,40)(17,41)(21,45,37,59)(22,46)(24,48,27,51)(26,50)(29,53)(30,55)(31,54)(36,58)(42,60)(43,61)(44,62)(52,63)(56,64);(1,5,6,23)(2,10,11,35)(3,15,16,22)(4,18,19,47)(7,27,13,24)(8,29,30,56)(9,32,33,57)(12,37,25,21)(14,39,40,46)(17,44,26,43)(20,51,38,48)(28,53,55,64)(31,52,36,42)(34,59,49,45)(41,62,50,61)(54,63,58,60)Polycyclic presentation:
2^9|a^2=efg,b^2=efh,c^2=ef,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361012);