Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13371 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #476 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1864 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
704 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,16,19,29)(8,21)(9,24)(10,27)(11,15)(12,18,30,25)(13,23)(14,22)(17,20,34,36)(26,32)(28,33,35,31);(5,7,8,18,10,20,23,35,11,19,9,25,22,36,26,28)(6,12,13,16,14,31,21,17,15,30,32,29,27,33,24,34);(2,4)(5,6)(7,17)(8,21)(9,24)(10,27)(11,15)(12,28)(13,23)(14,22)(16,20)(18,33)(19,34)(25,31)(26,32)(29,36)(30,35)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,f^b=fi,g^b=gi,d^c=dh,f^c=fhi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216647, C2×#148359, #72224878, … |
| maximal normal subgroup of | #72216647, C2×#148359, #72224878, … |
| maximal quotient of | #72216578, #72216581, C2×#148359 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55555);