Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2404 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1241 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7)(3,6,4,8)(9,10,14,20,15,13,16,19)(11,21,23,18,22,24,12,17);(1,3,5,4)(2,6,7,8)(9,11,15,22)(10,17,13,18)(12,14,23,16)(19,24,20,21);(2,7)(6,8)(9,12)(10,18)(11,14)(13,17)(15,23)(16,22)(19,21)(20,24);(1,4,5,3)(2,8,7,6)(9,13,15,10)(11,21,22,24)(12,17,23,18)(14,19,16,20)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,d^2=f,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,d^b=de,e^b=ei,e^c=ei,e^d=ei,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 359401);