Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11508 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | #1459 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
504 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 110 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)(5,6,10,16,12,17,24,36,11,18,28,29,9,15,19,35)(7,13,8,23,22,31,25,33,21,32,26,34,20,30,27,14);(1,2,3,4)(5,7,12,22,11,21,9,20)(6,13,17,31,18,32,15,30)(8,24,25,28,26,19,27,10)(14,16,23,36,33,29,34,35);(5,8)(6,14)(7,19)(9,27)(10,22)(11,26)(12,25)(13,29)(15,34)(16,32)(17,23)(18,33)(20,28)(21,24)(30,36)(31,35)Polycyclic presentation:
2^9|a^2=f,b^2=dh,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=ci,e^c=ei,f^c=fgi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72240000, #72240696, #72240697 |
| maximal normal subgroup of | #72240000, #72240696, #72240697 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41491);