Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C4×D8 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
312 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)(5,6,8,11,9,7,10,12)(14,20)(15,22)(16,21)(17,26)(18,24)(23,28)(25,27);(5,7,8,12,9,6,10,11)(13,14,21,25,18,22,26,28,19,23,17,15,24,27,16,20);(2,4)(13,15,16,23,24,28,17,22,19,25,26,14,18,20,21,27)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=d,d^2=gi,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,c^b=ci104 conjugacy classes — view the full character table.
| maximal subgroup of | #72286742, #72320165, #72320199 |
| maximal normal subgroup of | #72286742, #72320165, #72320199 |
| maximal quotient of | #72286915, #72320403 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47372);