Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13436 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2044 |
| commutator | C2×#41 | C22×C4 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
659 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 659 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(6,12)(7,16,19,34)(8,10)(13,27,29,36)(14,26)(15,22)(17,35,25,28)(18,33,23,32)(20,24)(30,31);(5,6,10,22,11,15,8,12)(7,17,23,29,19,25,18,13)(9,14,24,30,21,31,20,26)(16,27,33,28,34,36,32,35);(5,7,8,18,11,19,10,23)(6,13,12,25,15,29,22,17)(9,16,20,32,21,34,24,33)(14,28,26,27,31,35,30,36)Polycyclic presentation:
2^9|a^2=g,b^2=di,c^2=d,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fhi,c^b=cf,e^b=eh,f^b=fi,f^c=fi,g^c=gi,h^c=hi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72311847, #72311877, #72311884, … |
| maximal normal subgroup of | #72311847, #72311877, #72311884, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56024);