Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13402 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2034 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,11,17)(7,18)(8,23)(9,19,29,24)(10,16,12,32)(13,33)(14,31)(15,28,35,30)(20,27)(21,36)(22,25)(26,34);(5,7,9,14,12,22,30,34,11,21,29,33,10,20,28,23)(6,13,15,18,16,26,19,27,17,31,35,36,32,8,24,25);(2,4)(5,8)(6,14)(7,19)(9,27)(10,31)(11,26)(12,13)(15,20)(16,34)(17,33)(18,28)(21,24)(22,35)(23,32)(25,29)(30,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,f^b=fi,g^b=gi,d^c=di,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321300, #72321352, #72321394, … |
| maximal normal subgroup of | #72321300, #72321352, #72321394, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55506);