Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6374204 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2308 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62859 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4213 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)(3,13,16,8)(4,18,21,37)(5,22,24,10)(6,12)(7,11)(9,20,33,26)(14,39)(15,29)(17,36,25,30)(19,48)(23,52,55,34)(27,35)(28,42)(31,45)(32,49)(38,43)(40,61,62,60)(41,46)(44,57)(47,50)(51,54)(53,59)(56,63,58,64);(1,3)(2,8)(4,14)(5,15)(6,25)(7,16)(9,31)(10,29)(11,13)(12,36)(17,27)(18,46)(19,40)(20,45)(21,41)(22,38)(23,53)(24,43)(26,42)(28,33)(30,35)(32,58)(34,59)(37,39)(44,55)(47,61)(48,60)(49,64)(50,62)(51,56)(52,57)(54,63);(1,4,7,21)(2,9,11,33)(3,14,16,41)(5,19,24,50)(6,26,27,20)(8,28,13,31)(10,32,22,51)(12,37,35,18)(15,40,43,62)(17,45,25,42)(23,54,55,49)(29,56,38,58)(30,39,36,46)(34,48,52,47)(44,64,53,63)(57,60,59,61);(1,5)(2,10)(3,15)(4,19)(6,23)(7,24)(8,29)(9,32)(11,22)(12,34)(13,38)(14,40)(16,43)(17,44)(18,47)(20,49)(21,50)(25,53)(26,54)(27,55)(28,56)(30,57)(31,58)(33,51)(35,52)(36,59)(37,48)(39,60)(41,62)(42,63)(45,64)(46,61);(3,16)(8,13)(14,41)(15,43)(17,25)(28,31)(29,38)(30,36)(39,46)(40,62)(42,45)(44,53)(56,58)(57,59)(60,61)(63,64)Polycyclic presentation:
2^9|a^2=h,c^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,f^a=fi,c^b=cf,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,h^d=hi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281401);