Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,8,7)(9,10,13,14)(11,12,15,16)(17,18)(19,24)(20,23)(21,22);(9,11)(10,12)(13,15)(14,16)(18,21)(23,24);(6,7)(10,14)(11,15)(17,19,22,20)(18,23,21,24);(1,2)(3,4)(5,7,8,6)(9,11)(10,12)(13,15)(14,16)(17,20,22,19)(18,24,21,23);(9,12)(10,11)(13,16)(14,15)(17,21,22,18)(19,24,20,23)Polycyclic presentation:
2^9|a^2=fg,c^2=f,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
| maximal subgroup of | #72331387, #72331389, #72331428, … |
| maximal normal subgroup of | #72331387, #72331389, #72331428, … |
| maximal quotient of | #72333158 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4841949);