Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4426 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)(5,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,24,20,22);(1,2)(3,4)(10,11)(14,15)(17,19,21,20)(18,22,23,24);(1,2,3,4)(12,16)(14,15)(18,23)(22,24);(5,6)(7,8)(9,11,13,10)(12,15,16,14)(17,20,21,19)(18,24,23,22);(1,2)(3,4)(9,12)(10,14)(11,15)(13,16)(17,18)(19,22)(20,24)(21,23)Polycyclic presentation:
2^9|a^2=fgi,b^2=gh,c^2=h,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232467, #72232477, #72232551, … |
| maximal normal subgroup of | #72232467, #72232477, #72232551, … |
| maximal quotient of | #72278750, #72299040, #72299121, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2028882);