Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6374187 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2308 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62848 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4191 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,12,7,11,26,35)(3,13,17,36,16,8,24,29)(4,9,20,33,19,32,48,58)(5,21,22,37,23,10,25,34)(14,27,41,52,40,30,49,56)(15,28,43,53,42,38,44,54)(18,45,47,59,46,31,50,57)(39,55,61,64,60,51,62,63);(1,3)(2,8)(4,14)(5,15)(6,24)(7,16)(9,30)(10,28)(11,13)(12,36)(17,26)(18,39)(19,40)(20,49)(21,38)(22,44)(23,42)(25,43)(27,32)(29,35)(31,55)(33,52)(34,54)(37,53)(41,48)(45,51)(46,60)(47,62)(50,61)(56,58)(57,63)(59,64);(1,4)(2,9)(3,14)(5,18)(6,20)(7,19)(8,27)(10,31)(11,32)(12,33)(13,30)(15,39)(16,40)(17,41)(21,45)(22,47)(23,46)(24,49)(25,50)(26,48)(28,51)(29,52)(34,57)(35,58)(36,56)(37,59)(38,55)(42,60)(43,61)(44,62)(53,63)(54,64);(1,5,7,23)(2,10,11,21)(3,15,16,42)(4,18,19,46)(6,25,26,22)(8,28,13,38)(9,31,32,45)(12,37,35,34)(14,39,40,60)(17,44,24,43)(20,50,48,47)(27,51,30,55)(29,54,36,53)(33,59,58,57)(41,62,49,61)(52,64,56,63);(4,19)(9,32)(14,40)(18,46)(20,48)(27,30)(31,45)(33,58)(39,60)(41,49)(47,50)(51,55)(52,56)(57,59)(61,62)(63,64)Polycyclic presentation:
2^9|a^2=h,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,f^a=fi,c^b=cf,g^b=gi,h^b=hi,e^c=ei,h^d=hi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72310784 |
| maximal normal subgroup of | #72310784 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281392);