Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #442355 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15709 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
851 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)(3,13,17,36)(4,18,22,9)(5,23)(6,11)(7,12)(8,16,30,27)(10,26)(14,38)(15,43,44,34)(19,31)(20,32,51,46)(21,47,39,33)(24,29,54,56)(25,53)(28,41)(35,48)(37,42)(40,57)(45,52)(49,62)(50,58,60,63)(55,64)(59,61);(1,3,6,27,7,17,19,16)(2,8,11,36,12,30,31,13)(4,14,39,41,22,42,21,52)(5,24,25,15,26,54,48,44)(9,28,47,37,18,45,33,38)(10,34,35,29,23,43,53,56)(20,49,60,61,51,64,50,40)(32,57,63,62,46,59,58,55);(1,4,19,39,7,22,6,21)(2,9,31,47,12,18,11,33)(3,14,27,52,17,42,16,41)(5,20,48,60,26,51,25,50)(8,28,36,38,30,45,13,37)(10,32,53,63,23,46,35,58)(15,40,54,64,44,61,24,49)(29,55,43,59,56,62,34,57);(1,5,6,25,7,26,19,48)(2,10,31,53,12,23,11,35)(3,15,16,24,17,44,27,54)(4,20,21,50,22,51,39,60)(8,29,36,43,30,56,13,34)(9,32,47,63,18,46,33,58)(14,40,41,49,42,61,52,64)(28,55,38,59,45,62,37,57)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=ei,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 349552);