Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6408564 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #34364 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60648 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1735 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)(3,14,18,8)(4,19)(5,25,27,11)(6,10)(7,12)(9,23)(13,21)(15,41,43,33)(16,31)(17,32,28,30)(20,39,48,29)(22,51,53,46)(24,34)(26,38,50,37)(35,52,54,64)(36,49)(40,45)(42,59)(44,57)(47,56)(55,58)(60,63)(61,62);(1,3)(2,8)(4,20)(5,16)(6,28)(7,18)(9,33)(10,32)(11,31)(12,14)(13,30)(15,23)(17,21)(19,29)(22,47)(24,48)(25,40)(26,55)(27,45)(34,39)(35,59)(36,41)(37,58)(38,57)(42,52)(43,49)(44,50)(46,56)(51,60)(53,63)(54,61)(62,64);(1,4,21,49,7,24,6,23)(2,9,10,34,12,36,13,19)(3,15,28,48,18,43,17,20)(5,22,50,64,27,53,26,52)(8,29,30,41,14,39,32,33)(11,35,37,51,25,54,38,46)(16,42,55,63,45,62,44,47)(31,56,57,61,40,60,58,59);(2,10,12,13)(8,30,14,32)(9,34,36,19)(11,37,25,38)(29,41,39,33)(31,57,40,58)(35,51,54,46)(56,61,60,59);(1,5,7,27)(2,11,12,25)(3,16,18,45)(4,22,24,53)(6,26,21,50)(8,31,14,40)(9,35,36,54)(10,37,13,38)(15,42,43,62)(17,44,28,55)(19,46,34,51)(20,47,48,63)(23,52,49,64)(29,56,39,60)(30,57,32,58)(33,59,41,61)Polycyclic presentation:
2^9|a^2=h,c^2=gi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,c^b=cg,d^b=df,g^b=gi,h^b=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6315762);