Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11499 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C22 | C2×#342 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1718 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)(5,6,9,14,11,16,10,15)(7,17,19,32,21,34,20,33)(8,22,23,31,24,13,25,29)(12,26,27,18,28,35,30,36);(5,7,10,20,11,21,9,19)(6,12,15,27,16,28,14,30)(8,18,23,35,24,36,25,26)(13,17,29,33,22,34,31,32);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=g,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,f^b=fh,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72317357, #72317362, #72317386, … |
| maximal normal subgroup of | #72317357, #72317362, #72317386, … |
| maximal quotient of | C2×#146463 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53659);