Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7535046 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3549 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | #59816 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,15)(4,20)(5,26)(6,13)(7,11)(8,14)(9,18)(10,25)(12,27)(16,42)(17,45)(19,32)(21,35)(22,37)(23,36)(24,38)(28,40)(29,39)(30,34)(31,43)(33,47)(41,46)(44,63)(48,52)(49,57)(50,61)(51,60)(53,54)(55,58)(56,64)(59,62);(1,3,6,18,8,19,7,30)(2,9,11,15,14,34,13,32)(4,21,24,43,22,49,25,16)(5,17,27,46,29,47,28,55)(10,35,37,42,38,57,20,31)(12,33,26,41,40,45,39,58)(23,50,52,62,51,64,53,44)(36,59,54,61,60,63,48,56);(1,4,7,25,8,22,6,24)(2,10,13,20,14,38,11,37)(3,16,18,21,19,43,30,49)(5,23,28,53,29,51,27,52)(9,31,15,35,34,42,32,57)(12,36,39,48,40,60,26,54)(17,44,46,50,47,62,55,64)(33,56,41,59,45,61,58,63);(2,11,14,13)(4,22)(9,32,34,15)(10,20,38,37)(12,26,40,39)(16,43)(21,49)(23,51)(24,25)(31,35,42,57)(33,58,45,41)(36,48,60,54)(44,62)(50,64)(52,53)(56,59,61,63);(1,5,6,27,8,29,7,28)(2,12,11,26,14,40,13,39)(3,17,18,46,19,47,30,55)(4,23,24,52,22,51,25,53)(9,33,15,41,34,45,32,58)(10,36,37,54,38,60,20,48)(16,44,21,50,43,62,49,64)(31,56,35,59,42,61,57,63)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=h,e^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521282);