Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,11,7)(8,9,12,10)(13,14,16,19,18,22,17,21)(15,23,25,27,24,28,26,20);(1,2,3,4)(5,7,9,12,11,6,10,8)(14,19,22,21)(20,23,27,28);(5,8,10,6,11,12,9,7)(13,15,17,26,18,24,16,25)(14,20,19,23,22,27,21,28);(15,24)(20,27)(23,28)(25,26)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72215888, #72216052, #72216054, … |
| maximal normal subgroup of | #72215888, #72216052, #72216054, … |
| maximal quotient of | #72321836 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 111567);