Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7339802 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #48088 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23874 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1359 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)(3,14,18,39)(4,9)(5,10)(6,12)(7,13)(8,17,11,26)(15,43,19,42)(16,35,47,61)(20,32)(21,33)(22,34)(23,36,53,30)(24,37)(25,38)(27,40)(28,41)(29,45,31,48)(44,63,49,58)(46,56)(50,55,64,57)(51,59)(52,60)(54,62);(1,3,7,18)(2,8,13,11)(4,19,22,15)(5,23,25,53)(6,26,28,17)(9,31,34,29)(10,35,38,61)(12,39,41,14)(16,46,47,24)(20,50,52,64)(21,45,27,48)(30,56,36,37)(32,58,60,63)(33,43,40,42)(44,51,49,54)(55,59,57,62);(1,4,7,22)(2,9,13,34)(3,15,18,19)(5,20,25,52)(6,27,28,21)(8,29,11,31)(10,32,38,60)(12,40,41,33)(14,42,39,43)(16,44,47,49)(17,48,26,45)(23,50,53,64)(24,54,46,51)(30,55,36,57)(35,58,61,63)(37,62,56,59);(1,5,6,24,7,25,28,46)(2,10,12,37,13,38,41,56)(3,16,26,53,18,47,17,23)(4,20,21,51,22,52,27,54)(8,30,39,61,11,36,14,35)(9,32,33,59,34,60,40,62)(15,44,48,50,19,49,45,64)(29,55,42,58,31,57,43,63);(1,3)(2,11)(4,15)(5,16)(6,26)(7,18)(8,13)(9,31)(10,36)(12,14)(17,28)(19,22)(20,44)(21,48)(23,46)(24,53)(25,47)(27,45)(29,34)(30,38)(32,57)(33,43)(35,37)(39,41)(40,42)(49,52)(50,51)(54,64)(55,60)(56,61)(58,62)(59,63)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=fg,f^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ch,d^b=dfh,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,h^d=hi,f^e=fi,g^e=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7247002);