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, socle | C22 | #1864 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
976 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6,10,17,12,18,11,29)(7,19,21,33,23,30,22,34)(8,24,27,16,9,13,25,15)(14,26,31,20,32,28,35,36);(6,13,18,24)(8,25)(9,27)(10,11)(14,30,32,19)(15,17,16,29)(20,26)(21,22)(28,36)(31,33,35,34);(5,7)(6,14)(8,26)(9,28)(10,22)(11,21)(12,23)(13,30)(15,33)(16,34)(17,35)(18,32)(19,24)(20,25)(27,36)(29,31)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,e^2=h,f^2=i,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 | #72231484, #72231513, #72231517, … |
| maximal normal subgroup of | #72231484, #72231513, #72231517, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53957);