Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454426 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23387 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,3)(5,6)(7,8)(9,10,15,21)(11,23,24,13)(12,19,17,18)(14,22,16,20);(1,3,4,2)(9,11,16,18,15,24,14,19)(10,17,22,13,21,12,20,23);(2,3)(9,12)(10,18)(11,22)(13,16)(14,23)(15,17)(19,21)(20,24);(6,8)(9,13,16,12,15,23,14,17)(10,19,22,11,21,18,20,24)Polycyclic presentation:
2^9|a^2=f,b^2=ef,d^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364431);