Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468068 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16600 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1058 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)(5,10)(6,25)(7,12)(8,16)(9,21)(11,28)(14,43)(15,47)(17,42)(19,33)(20,34)(22,57)(23,39)(24,37)(26,38)(27,36)(29,45)(30,49)(31,53)(32,51)(35,56)(40,46)(41,50)(44,62)(48,61)(52,60)(54,55)(58,64)(59,63);(1,3)(2,8,12,31)(4,14)(5,15)(6,16)(7,17)(9,32,18,29)(10,30,37,60)(11,13,25,42)(19,44)(20,45)(21,46)(22,48)(23,49)(24,50)(26,51)(27,52)(28,53)(33,61,54,58)(34,40,38,43)(35,62,57,59)(36,47,39,41)(55,63)(56,64);(1,4)(2,9,12,18)(3,14)(5,22)(6,26)(7,21)(8,29,31,32)(10,35,37,57)(11,38,25,34)(13,40,42,43)(15,48)(16,51)(17,46)(19,23)(20,28)(24,56)(27,55)(30,59,60,62)(33,39,54,36)(41,61,47,58)(44,49)(45,53)(50,64)(52,63);(1,5,7,24)(2,10,12,37)(3,15,17,50)(4,19,21,55)(6,27,28,23)(8,30,31,60)(9,33,18,54)(11,39,25,36)(13,41,42,47)(14,44,46,63)(16,52,53,49)(20,56,26,22)(29,58,32,61)(34,57,38,35)(40,59,43,62)(45,64,51,48)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 375265);