Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384326 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14377 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
965 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)(3,14,19,39)(4,20)(5,26,30,11)(6,12)(7,31)(8,13)(9,17,34,18)(10,24)(15,41,21,32)(16,45)(22,52,56,49)(23,36)(25,50)(27,33)(28,58,29,38)(35,55,57,54)(37,47)(40,43,48,44)(42,60)(46,61)(51,63)(53,64)(59,62);(1,3)(2,9)(4,21)(5,27)(6,18)(7,17)(8,19)(10,32)(11,37)(12,39)(13,34)(14,31)(15,25)(16,28)(20,48)(22,53)(23,43)(24,44)(26,45)(29,47)(30,46)(33,38)(35,62)(36,41)(40,50)(42,55)(49,60)(51,54)(52,63)(56,59)(57,64)(58,61);(1,4,7,24,8,25,6,23)(2,10,12,20,13,36,31,50)(3,15,18,44,19,21,17,43)(5,22,29,55,30,56,28,54)(9,32,14,40,34,41,39,48)(11,35,38,49,26,57,58,52)(16,42,46,53,47,51,27,59)(33,60,37,62,61,63,45,64);(1,5,6,28,8,30,7,29)(2,11,12,38,13,26,31,58)(3,16,17,27,19,47,18,46)(4,22,23,54,25,56,24,55)(9,33,14,37,34,61,39,45)(10,35,20,49,36,57,50,52)(15,42,43,59,21,51,44,53)(32,60,40,62,41,63,48,64)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,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, 291522);