Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384163 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14373 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1035 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,11,27,36)(3,13,17,37,16,8,25,30)(4,9,21,33,20,32,50,58)(5,22,23,38,24,10,26,35)(14,28,41,53,18,39,46,59)(15,34,43,55,42,29,44,54)(19,47,49,60,48,31,51,57)(40,52,61,63,45,56,62,64);(1,3)(2,8)(4,18)(5,15)(6,25)(7,16)(9,28)(10,34)(11,13)(12,37)(14,20)(17,27)(19,45)(21,41)(22,29)(23,44)(24,42)(26,43)(30,36)(31,56)(32,39)(33,59)(35,54)(38,55)(40,48)(46,50)(47,52)(49,61)(51,62)(53,58)(57,63)(60,64);(1,4)(2,9)(3,14,16,18)(5,19)(6,21)(7,20)(8,28,13,39)(10,31)(11,32)(12,33)(15,40,42,45)(17,41,25,46)(22,47)(23,49)(24,48)(26,51)(27,50)(29,52,34,56)(30,53,37,59)(35,57)(36,58)(38,60)(43,61,44,62)(54,63,55,64);(1,5,7,24)(2,10,11,22)(3,15,16,42)(4,19,20,48)(6,26,27,23)(8,29,13,34)(9,31,32,47)(12,38,36,35)(14,40,18,45)(17,44,25,43)(21,51,50,49)(28,52,39,56)(30,55,37,54)(33,60,58,57)(41,62,46,61)(53,64,59,63)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,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, 291361);