Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#576 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(10,15)(12,13)(16,22)(17,21)(18,20)(23,24);(9,10)(11,16)(12,15)(13,21)(14,17)(18,22)(19,23)(20,24);(2,4)(9,11)(10,16)(12,18)(13,20)(14,19)(15,22)(17,23)(21,24);(9,12)(11,18)(13,14)(15,21)(19,20)(22,24);(11,19)(16,23)(18,20)(22,24)Polycyclic presentation:
2^9|a^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,d^b=df,f^b=fi,e^c=ei,f^d=fi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72240755, #72240843, #72249594, … |
| maximal normal subgroup of | #72240755, #72240843, #72249594, … |
| maximal quotient of | #72240784, #72264795, #72264816, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286357);