Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12948 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1864 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #10788 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
959 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,15,19,38)(4,21,23,57)(5,25)(6,30)(7,34)(8,13)(9,17,39,18)(10,22,41,58)(11,29)(12,33)(14,28)(16,50)(20,45,35,43)(24,46,36,44)(26,40,31,47)(27,42,32,48)(37,53)(49,59)(51,62)(52,61)(54,64)(55,60)(56,63);(1,3)(2,9,13,39)(4,16)(5,26)(6,31)(7,35)(8,19)(10,37,41,50)(11,43,12,45)(14,47,34,40)(15,30,38,25)(17,29)(18,33)(20,28)(21,49,57,62)(22,54)(23,53)(24,56)(27,51)(32,59)(36,52)(42,63,48,61)(44,60,46,64)(55,58);(1,4)(2,10)(3,16)(5,27)(6,32)(7,36)(8,23)(9,37)(11,44)(12,46)(13,41)(14,48)(15,49)(17,54)(18,55)(19,53)(20,56)(21,25)(22,29)(24,28)(26,51)(30,57)(31,59)(33,58)(34,42)(35,52)(38,62)(39,50)(40,63)(43,60)(45,64)(47,61)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,d^b=dh,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280192, #72280652, #72281111, … |
| maximal normal subgroup of | #72280192, #72280652, #72281111, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53955);