Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,7,8)(9,10)(11,15)(12,14)(13,16)(18,20)(19,21)(23,24);(1,2)(3,4)(17,18)(19,20)(21,24)(22,23);(2,4)(9,11,12,16)(10,13,14,15)(17,19,22,21)(18,20,23,24);(10,14)(13,15)(17,20,19,18,22,24,21,23)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=ei,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72240765, #72240767, #72240778, … |
| maximal normal subgroup of | #72240765, #72240767, #72240778, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291547);