Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419458 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2709 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15730 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
773 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,13,6,27)(3,14,17,9,18,39,28,33)(4,10,30,35,22,36,21,53)(5,23,25,38,26,11,29,40)(15,43,45,58,19,41,49,31)(16,37,47,59,46,42,24,32)(20,34,51,62,52,55,54,61)(44,60,50,64,48,63,56,57);(1,3,8,17,7,18,6,28)(2,9,12,39,13,33,27,14)(4,19,30,49,22,15,21,45)(5,24,25,16,26,47,29,46)(10,31,35,43,36,58,53,41)(11,37,40,59,23,42,38,32)(20,50,51,48,52,56,54,44)(34,60,62,64,55,63,61,57);(1,4)(2,10)(3,15)(5,20)(6,21)(7,22)(8,30)(9,31)(11,34)(12,35)(13,36)(14,41)(16,44)(17,45)(18,19)(23,55)(24,56)(25,51)(26,52)(27,53)(28,49)(29,54)(32,57)(33,58)(37,60)(38,61)(39,43)(40,62)(42,63)(46,48)(47,50)(59,64);(1,5,6,29,7,26,8,25)(2,11,12,40,13,23,27,38)(3,16,28,24,18,46,17,47)(4,20,21,54,22,52,30,51)(9,32,39,37,33,59,14,42)(10,34,35,62,36,55,53,61)(15,44,49,56,19,48,45,50)(31,57,43,60,58,64,41,63)Polycyclic presentation:
2^9|a^2=efg,b^2=efg,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 349950);