Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #647656 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6,7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,22)(20,21)(23,24);(2,4)(10,13)(14,15)(18,21)(19,23);(1,2)(3,4)(5,6)(7,8)(17,19,20,23)(18,22,21,24);(9,11)(10,14)(12,16)(13,15);(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,24)Polycyclic presentation:
2^9|a^2=g,c^2=fi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | #72335431, #72335437, #72335472, … |
| maximal normal subgroup of | #72335431, #72335437, #72335472, … |
| maximal quotient of | #72335674, C2×#663633 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 570829);