Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13357 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C22 | #1867 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
817 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)(5,6,11,16,12,17,10,15)(7,18,21,29,22,36,27,35)(8,14,24,32,9,25,26,34)(13,28,30,23,31,19,33,20);(5,7,9,20,12,22,8,23)(6,13,14,18,17,31,25,36)(10,27,24,19,11,21,26,28)(15,33,34,35,16,30,32,29);(2,4)(5,8)(6,14)(7,19)(9,12)(10,26)(11,24)(13,29)(15,34)(16,32)(17,25)(18,33)(20,21)(22,28)(23,27)(30,36)(31,35)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fhi,c^b=cf,f^b=fhi,h^b=hi,d^c=dh,f^c=fi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72250688, #72257229, #72257236, … |
| maximal normal subgroup of | #72250688, #72257229, #72257236, … |
| maximal quotient of | #72265255 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55459);