Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
7338 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)(3,4)(5,6,7,8)(9,10)(11,16)(12,14)(13,15)(17,18)(19,23)(20,22)(21,24);(9,11,13,12)(10,14,15,16)(17,19,20,24)(18,21,22,23);(2,4)(19,24)(21,23);(10,15)(14,16)(18,22)(21,23);(9,12,13,11)(10,14,15,16)Polycyclic presentation:
2^9|a^2=i,b^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=df,c^b=ch152 conjugacy classes — view the full character table.
| maximal subgroup of | #72237121, #72237143, #72237154, … |
| maximal normal subgroup of | #72237121, #72237143, #72237154, … |
| maximal quotient of | #72239720, #72239725, #72239887, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 421406);