Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#585 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
16066 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)(5,6,7,8)(10,15)(12,13)(16,23)(17,20)(18,21)(22,24);(9,10)(11,16)(12,15)(13,20)(14,17)(18,22)(19,24)(21,23);(9,11)(10,16)(12,18)(13,21)(14,19)(15,22)(17,24)(20,23);(2,4)(9,12,14,13)(10,15,17,20)(11,18,19,21)(16,22,24,23)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,e^b=ei,f^b=fi,e^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72258252, #72259189, #72259221, … |
| maximal normal subgroup of | #72258252, #72259189, #72259221, … |
| maximal quotient of | #72317089, #72317090 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299326);