Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(9,10,12,13)(11,15,16,14)(17,18,22,24)(19,20,21,23);(6,8)(9,10)(11,15)(12,13)(14,16)(17,19,22,21)(18,20,24,23);(5,6,7,8)(10,13)(14,15)(17,20,22,23)(18,19,24,21);(9,10)(11,14)(12,13)(15,16)(17,18)(19,20)(21,23)(22,24);(1,3,4,2)(9,11)(10,14)(12,16)(13,15)(17,21,22,19)(18,23,24,20)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=gh,e^2=fi,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=df116 conjugacy classes — view the full character table.
| maximal subgroup of | #72328201, #72328211, #72328267, … |
| maximal normal subgroup of | #72328201, #72328211, #72328267, … |
| maximal quotient of | #72331449, #72333319 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 447820);