Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #372790 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14100 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1388 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)(3,8,15,31)(4,17)(5,10,23,35)(6,25)(7,11)(9,21)(12,22)(13,38,18,49)(14,41)(16,45,42,30)(19,51,53,48)(20,36)(24,37,27,34)(26,33)(28,46,56,40)(29,44)(32,54,59,52)(39,60)(43,57)(47,58)(50,64)(55,62)(61,63);(1,3)(2,8,11,31)(4,18,20,13)(5,14)(6,16)(7,15)(9,28)(10,29,35,41)(12,30,25,45)(17,38)(19,50,53,39)(21,46,26,40)(22,42)(23,44)(24,43)(27,47)(32,55)(33,56)(34,57,37,58)(36,49)(48,60)(51,64)(52,62,54,61)(59,63);(1,4)(2,9)(3,13)(5,19)(6,26)(7,20)(8,28)(10,32)(11,33)(12,36)(14,39)(15,18)(16,46)(17,25)(21,22)(23,53)(24,54)(27,52)(29,55)(30,49)(31,56)(34,48)(35,59)(37,51)(38,45)(40,42)(41,63)(43,62)(44,50)(47,61)(57,60)(58,64);(1,5,22,24,7,23,6,27)(2,10,25,34,11,35,12,37)(3,14,42,43,15,44,16,47)(4,19,21,54,20,53,26,52)(8,29,45,57,31,41,30,58)(9,32,17,48,33,59,36,51)(13,39,40,62,18,50,46,61)(28,55,38,60,56,63,49,64)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=efgi,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279987);