Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #469875 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2607 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #17034 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
579 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)(4,9)(5,20,22,49)(6,12)(7,11)(10,23,35,52)(13,26)(14,41,43,61)(15,29)(16,28)(17,47,21,50)(18,32)(19,33)(24,37)(25,36)(27,44,56,64)(30,39)(31,48,34,51)(38,59,42,62)(40,54)(45,58)(46,57)(53,60,55,63);(1,3,6,15,7,16,25,46)(2,8,12,29,11,28,36,57)(4,13,24,45,18,39,19,40)(5,14,23,44,22,43,52,64)(9,30,37,54,32,26,33,58)(10,27,20,41,35,56,49,61)(17,38,51,63,21,42,48,60)(31,55,50,59,34,53,47,62);(1,4)(2,9)(3,13)(5,21,22,17)(6,24)(7,18)(8,26)(10,34,35,31)(11,32)(12,37)(14,42,43,38)(15,45)(16,39)(19,25)(20,47,49,50)(23,48,52,51)(27,55,56,53)(28,30)(29,58)(33,36)(40,46)(41,59,61,62)(44,60,64,63)(54,57);(1,5)(2,10,11,35)(3,14)(4,17)(6,23)(7,22)(8,27,28,56)(9,31,32,34)(12,20,36,49)(13,38)(15,44)(16,43)(18,21)(19,48)(24,51)(25,52)(26,53,30,55)(29,41,57,61)(33,47,37,50)(39,42)(40,60)(45,63)(46,64)(54,59,58,62)Polycyclic presentation:
2^9|a^2=f,b^2=eh,c^2=f,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^c=de,e^c=ei,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 382425);