Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12707 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1744 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 1744 conjugacy classes of subgroups — too many to draw. Only its 202 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:
(5,6,9,13,10,14,24,30)(7,15,18,32,17,31,22,35)(8,19,21,29,20,12,23,28)(11,25,27,33,26,34,36,16);(1,2,3,4)(5,7)(6,11,14,26)(8,16)(9,22)(10,17)(12,15,19,31)(13,27,30,36)(18,24)(20,33)(21,34)(23,25)(28,32,29,35);(8,20)(12,19)(15,31)(16,33)(21,23)(25,34)(28,29)(32,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,e^b=ei,f^b=fhi,d^c=di,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286236, #72286317, #72286811, … |
| maximal normal subgroup of | #72286236, #72286317, #72286811, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53677);