Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10)(11,14)(12,13)(15,16)(17,18,20,22)(19,23,24,21);(5,6)(7,8)(10,13)(14,16);(2,4)(6,8);(17,19,20,24)(18,21,22,23);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch128 conjugacy classes — view the full character table.
| maximal subgroup of | #72264460, #72264469, #72264477, … |
| maximal normal subgroup of | #72264460, #72264469, #72264477, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 515655);