Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1501 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)(10,14)(11,17,18,20)(15,23,21,24)(16,22);(2,4)(6,8)(9,10,13,16,12,14,19,22)(11,15,20,23,18,21,17,24);(5,6,7,8)(9,11)(10,15)(12,18)(13,20)(14,21)(16,23)(17,19)(22,24);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,f^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72309747, D4×#1938, #72310137 |
| maximal normal subgroup of | #72309747, D4×#1938, #72310137 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275644);