Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7339 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1316 |
| commutator | C22×C4 | C4×C8 |
| Frattini, cosocle | C2×#381 | C22 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C25 ◁ C2×#381 ◁ G | |
| LECS | G ▷ C2×#381 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
462 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 462 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 64 |
A minimal faithful permutation representation — 2 generators on 36 points:
(5,6,9,13)(7,15,17,33,18,28,34,36)(8,19,21,31)(10,14,23,32)(11,24,25,20,26,35,30,16)(12,27,29,22);(1,2,3,4)(5,7)(6,11)(8,20)(9,17)(10,18)(12,28)(13,30)(14,26)(15,19)(16,22)(21,35)(23,34)(24,27)(25,32)(29,33)(31,36)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=g,b^a=bc,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=dfi,g^b=gi,d^c=dh,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72267758, #72267777, #72285529, … |
| maximal normal subgroup of | #72267758, #72267777, #72285529, … |
| maximal quotient of | #72267760 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1270);