Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10564 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #394 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1440 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,3)(4,6,8,7)(9,10,14,18,13,17,21,24)(11,15,19,22,12,16,20,23);(1,3,5,2)(4,7,8,6)(9,11,13,12)(10,15,17,16)(14,20,21,19)(18,23,24,22);(2,6)(3,7)(4,8)(9,12,13,11)(10,16)(14,19,21,20)(15,17)(18,22)(23,24)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=d,e^2=hi,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,e^b=ei,f^b=fhi,g^b=gh,e^c=ei,f^c=fhi,g^c=gh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 27092);