Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11522 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | #1464 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
520 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 520 conjugacy classes of subgroups — too many to draw. Only its 114 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:
(1,2,4,3)(5,6)(7,11,16,23)(8,13)(9,14)(10,12)(15,22,18,25)(17,32,31,24)(19,26)(20,28)(21,27)(29,34)(30,36,33,35);(5,7,8,17,9,18,20,33,10,16,19,31,21,15,29,30)(6,11,13,24,14,25,28,36,12,23,26,32,27,22,34,35);(1,3,4,2)(7,15,16,18)(8,19)(11,22,23,25)(13,26)(17,33,31,30)(20,29)(24,36,32,35)(28,34)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,f^2=h,h^2=i,b^a=bd,d^a=di,f^a=fg,c^b=ch,d^b=dg,e^b=ei,g^b=gi,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72240697, #72240698, #72240699, … |
| maximal normal subgroup of | #72240697, #72240698, #72240699, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35574);