Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #385905 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14417 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1358 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)(3,14,19,9)(4,20,24,37)(5,25)(6,13)(7,30)(8,12)(10,23,36,28)(11,26)(15,34)(16,45,47,56)(17,41,18,33)(21,31)(22,52,55,35)(27,40)(29,39)(32,46,38,49)(42,48)(43,58)(44,59)(50,54,57,53)(51,61)(60,63)(62,64);(1,3)(2,9)(4,21)(5,16)(6,18)(7,17)(8,19)(10,34)(11,38)(12,14)(13,41)(15,23)(20,42)(22,51)(24,48)(25,56)(26,49)(27,47)(28,44)(29,46)(30,33)(31,37)(32,39)(35,61)(36,59)(40,45)(43,54)(50,58)(52,60)(53,64)(55,63)(57,62);(1,4,7,23,8,24,6,28)(2,10,30,20,12,36,13,37)(3,15,18,21,19,44,17,48)(5,22,26,54,27,55,29,53)(9,31,41,59,14,42,33,34)(11,35,40,50,39,52,25,57)(16,43,46,51,47,64,49,63)(32,58,45,61,38,62,56,60);(1,5)(2,11)(3,16)(4,22)(6,29)(7,26)(8,27)(9,32)(10,35)(12,39)(13,25)(14,38)(15,43)(17,49)(18,46)(19,47)(20,50)(21,51)(23,54)(24,55)(28,53)(30,40)(31,58)(33,56)(34,60)(36,52)(37,57)(41,45)(42,62)(44,64)(48,63)(59,61)Polycyclic presentation:
2^9|a^2=h,c^2=eg,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,f^c=fi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293101);