Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13357 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C22 | #1889 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1364 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 1364 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) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,12,16,9,15,11,27)(7,17,19,29,10,13,26,30)(8,20,24,32,22,28,23,36)(14,31,34,18,33,21,35,25);(1,2,3,4)(5,7,9,10)(6,13,15,17)(8,21,22,31)(11,26,12,19)(14,32,33,36)(16,30,27,29)(18,24,25,23)(20,34,28,35);(5,8)(6,14)(7,18)(9,22)(10,25)(11,24)(12,23)(13,28)(15,33)(16,35)(17,20)(19,31)(21,26)(27,34)(29,36)(30,32)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,e^b=eh,f^b=fi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^e=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72257211, #72257231, #72257236, … |
| maximal normal subgroup of | #72257211, #72257231, #72257236, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55370);