Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9471 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1436 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(5,8)(9,10,13,17,14,18,12,15)(11,16,21,23,19,24,20,22);(1,2)(3,5)(4,6)(7,8)(10,15,18,17)(11,19)(16,23,24,22)(20,21);(1,3)(2,5)(4,7)(6,8)(9,11,13,21,14,19,12,20)(10,16,17,23,18,24,15,22)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,c^b=ch,e^b=ei,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17101);