Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396100 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2414 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15219 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1277 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)(3,15,18,9)(4,10)(5,24)(6,14)(7,13)(8,12)(11,26)(16,30,44,41)(17,46,47,54)(19,32,28,39)(20,49,45,33)(21,51)(22,40)(23,35)(25,37)(27,38)(29,36)(31,48,42,55)(34,52)(43,62,63,58)(50,61,64,57)(53,60)(56,59);(1,3,7,19,8,18,6,28)(2,9,13,39,12,15,14,32)(4,20,22,16,23,45,29,44)(5,17,26,48,27,47,25,55)(10,33,40,41,35,49,36,30)(11,31,38,46,37,42,24,54)(21,50,52,43,53,64,56,63)(34,58,60,61,59,62,51,57);(1,4)(2,10)(3,16)(5,21)(6,29)(7,22)(8,23)(9,30)(11,34)(12,35)(13,40)(14,36)(15,41)(17,43)(18,44)(19,45)(20,28)(24,51)(25,56)(26,52)(27,53)(31,57)(32,49)(33,39)(37,59)(38,60)(42,61)(46,58)(47,63)(48,64)(50,55)(54,62);(1,5,7,26,8,27,6,25)(2,11,14,24,12,37,13,38)(3,17,19,48,18,47,28,55)(4,21,22,52,23,53,29,56)(9,31,32,54,15,42,39,46)(10,34,36,51,35,59,40,60)(16,43,45,64,44,63,20,50)(30,57,49,62,41,61,33,58)Polycyclic presentation:
2^9|a^2=f,b^2=eh,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 323396);