Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24808 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2416 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13974 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
941 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)(4,20,24,37)(5,25)(6,29)(7,12)(8,13)(9,17)(10,23,38,30)(11,26)(15,31,43,57)(16,44)(18,34)(19,33)(21,35,42,56)(22,36,51,60)(27,39)(28,53)(32,47)(40,45)(41,59,62,64)(46,58)(48,54,52,50)(49,63,61,55);(1,3)(2,9,13,34)(4,21)(5,16)(6,17)(7,18)(8,19)(10,35,38,56)(11,32,39,44)(12,14,29,33)(15,23)(20,31,37,57)(22,49)(24,42)(25,40,53,58)(26,45)(27,46)(28,47)(30,43)(36,59,60,64)(41,50)(48,55,52,63)(51,61)(54,62);(1,4)(2,10)(3,15)(5,22)(6,23)(7,30)(8,24)(9,31)(11,36)(12,37)(13,38)(14,35)(16,41)(17,42)(18,21)(19,43)(20,29)(25,52)(26,50)(27,54)(28,51)(32,55)(33,56)(34,57)(39,60)(40,59)(44,63)(45,61)(46,49)(47,62)(48,53)(58,64);(1,5,6,26,8,28,7,27)(2,11,12,25,13,39,29,53)(3,16,17,45,19,47,18,46)(4,22,23,50,24,51,30,54)(9,32,14,40,34,44,33,58)(10,36,37,52,38,60,20,48)(15,41,42,61,43,62,21,49)(31,55,35,59,57,63,56,64)Polycyclic presentation:
2^9|a^2=h,b^2=efh,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272523);