Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #635984 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,12,14)(11,15,16,13)(17,18,21,24)(19,22,23,20);(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,23)(18,22,24,20);(5,6)(7,8)(9,11,12,16)(10,13,14,15)(17,20,21,22)(18,23,24,19);(1,2)(3,4)(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,23)(21,24);(9,10)(11,13)(12,14)(15,16)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72281062, #72281346, #72281883, … |
| maximal normal subgroup of | #72281062, #72281346, #72281883, … |
| maximal quotient of | #72332502 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 543242);