Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #398434 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14762 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1532 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)(4,9)(5,22)(6,25)(7,12)(8,16)(10,24)(11,26)(14,37)(15,41)(17,36)(18,34)(19,47)(20,50)(21,32)(23,33)(27,46)(28,56)(29,44)(30,49)(31,38)(35,43)(39,58)(40,55)(42,51)(45,59)(48,57)(52,53)(54,61)(60,64)(62,63);(1,3,6,16,7,17,26,44)(2,8,11,13,12,29,25,36)(4,18,20,46,21,14,38,40)(5,15,23,42,24,43,53,56)(9,27,31,34,32,55,50,37)(10,28,33,35,22,51,52,41)(19,45,48,62,49,39,60,61)(30,54,57,58,47,63,64,59);(1,4,7,21)(2,9,12,32)(3,14)(5,19,24,49)(6,20,26,38)(8,27)(10,30,22,47)(11,31,25,50)(13,34)(15,39)(16,40)(17,18)(23,48,53,60)(28,54)(29,55)(33,57,52,64)(35,58)(36,37)(41,59)(42,61)(43,45)(44,46)(51,63)(56,62);(1,5,6,23,7,24,26,53)(2,10,11,33,12,22,25,52)(3,15,16,42,17,43,44,56)(4,19,20,48,21,49,38,60)(8,28,13,35,29,51,36,41)(9,30,31,57,32,47,50,64)(14,39,40,61,18,45,46,62)(27,54,34,58,55,63,37,59)Polycyclic presentation:
2^9|b^2=e,c^2=fi,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305631);