Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,3)(9,10,14,18,15,11,13,17)(12,20,23,19,21,24,22,16);(5,6,7,8)(9,11,15,10)(12,19,21,16)(13,17,14,18)(20,22,24,23);(9,12,13,22,15,21,14,23)(10,16,18,20,11,19,17,24);(1,3,4,2)(12,21)(16,19)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,d^2=eg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,e^b=ei,g^b=gi,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72321505, #72321536, #72321621, … |
| maximal normal subgroup of | #72321505, #72321536, #72321621, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261881);