Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11541 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
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 | |
1744 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)(5,6,9,13)(7,15,17,32)(8,19,21,30)(10,14,25,31)(11,26,27,16)(12,24,29,22)(18,33,23,35)(20,28,34,36);(5,7)(6,11,14,28)(8,20,22,16)(9,23,25,17)(10,18)(12,15)(13,27)(19,33)(21,26)(24,34)(29,32,30,35)(31,36);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=g,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,d^b=dh,f^b=fh,h^b=hi,f^d=fi,g^d=gi,g^f=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72267924, #72267928, #72280625, … |
| maximal normal subgroup of | #72267924, #72267928, #72280625, … |
| maximal quotient of | #72268051 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53759);