Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D8 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D16 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14)(15,22)(16,26)(17,20)(18,21)(19,24)(25,27);(2,4)(13,15)(14,19)(16,23)(17,27)(18,25)(20,28)(21,24)(22,26);(6,8)(14,20)(15,23)(16,21)(17,26)(19,27)(22,24)(25,28)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,h^a=hi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,h^b=hi,d^c=dh,e^c=ei,f^c=fh,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72322385, #72322452, #72322502, … |
| maximal normal subgroup of | #72322385, #72322452, #72322502, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55104);