Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419786 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15273 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23414 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1553 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)(3,15,19,39)(4,10)(5,23,26,41)(6,12)(7,14)(8,13)(9,18,32,27)(11,25,38,29)(16,43,46,42)(17,31)(20,50,53,55)(21,35)(22,36)(24,37)(28,40)(30,47,33,45)(34,54,61,52)(44,56)(48,58)(49,57)(51,62)(59,63)(60,64);(1,3)(2,9)(4,16)(5,24)(6,27)(7,18)(8,19)(10,33)(11,37)(12,39)(13,32)(14,15)(17,25)(20,51)(21,47)(22,46)(23,31)(26,48)(28,45)(29,49)(30,36)(34,60)(35,42)(38,58)(40,43)(41,57)(44,52)(50,56)(53,64)(54,63)(55,59)(61,62);(1,4)(2,10)(3,16)(5,20)(6,21)(7,28)(8,22)(9,30)(11,34)(12,35)(13,36)(14,40)(15,42)(17,44)(18,45)(19,46)(23,55)(24,51)(25,52)(26,53)(27,47)(29,54)(31,56)(32,33)(37,62)(38,61)(39,43)(41,50)(48,64)(49,63)(57,59)(58,60);(1,5,6,25,8,26,7,29)(2,11,12,23,13,38,14,41)(3,17,18,24,19,49,27,48)(4,20,21,52,22,53,28,54)(9,31,15,37,32,57,39,58)(10,34,35,55,36,61,40,50)(16,44,45,51,46,63,47,64)(30,56,42,62,33,59,43,60)Polycyclic presentation:
2^9|a^2=f,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72285925, #72286066, #72286206, … |
| maximal normal subgroup of | #72285925, #72286066, #72286206, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326982);