Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13404 |
| 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 | |
664 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)(5,6,11,15)(7,18)(8,22)(9,27,30,23)(10,16,31,17)(12,32)(13,26)(14,33,34,29)(19,25)(20,35)(21,24)(28,36);(5,7,9,19,10,21,29,13,11,20,30,32,31,36,33,22)(6,12,14,18,17,26,27,28,15,25,34,35,16,8,23,24);(2,4)(5,8)(6,13)(7,14)(9,28)(10,12)(11,26)(15,22)(16,19)(17,32)(18,33)(20,34)(21,27)(23,36)(24,30)(25,31)(29,35)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,e^b=ei,g^b=gi,d^c=di,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321300, #72321321, #72321333, … |
| maximal normal subgroup of | #72321300, #72321321, #72321333, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55500);