Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1324 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)(3,4)(5,6,8,7)(9,10,14,18,13,16,12,17)(11,19,23,15,22,20,21,24);(9,11,12,23,13,22,14,21)(10,15,17,20,16,24,18,19);(2,4)(5,6,8,7)(10,16)(15,24)(17,18)(19,20);(5,7,8,6)(9,10,14,18,13,16,12,17)(11,20,23,24,22,19,21,15)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=eg,d^2=g,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^b=de,e^b=ei,g^b=gi,d^c=di116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263917);