Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13350 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1387 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 1387 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)(6,11)(8,19)(12,27)(13,25)(14,26)(15,32)(16,33)(17,22)(20,30)(21,34)(24,29)(31,36);(5,6,8,11,10,13,19,25)(7,15,16,32,18,31,33,36)(9,21,20,34,23,14,30,26)(12,28,24,22,29,35,27,17);(5,7)(6,12)(8,16)(9,22)(10,18)(11,24)(13,29)(14,31)(15,21)(17,23)(19,33)(20,35)(25,27)(26,36)(28,30)(32,34)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^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,h^c=hi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72311847, #72311876, #72311880, … |
| maximal normal subgroup of | #72311847, #72311876, #72311880, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55374);