Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| 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 | |
4461 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(6,7)(9,10)(11,14)(12,15)(13,16)(17,18)(19,20)(21,22)(23,24);(9,11)(10,13)(12,16)(14,15)(17,19,23,22)(18,21,24,20);(1,2,4,3)(5,6,8,7)(17,20,23,21)(18,19,24,22);(5,6,8,7)(9,11)(10,14)(12,16)(13,15)(17,21,23,20)(18,19,24,22);(1,3,4,2)(5,7,8,6)(9,10)(11,13)(12,15)(14,16)(17,22,23,19)(18,21,24,20)Polycyclic presentation:
2^9|b^2=h,c^2=gh,d^2=i,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72304050, #72304082, #72304104, … |
| maximal normal subgroup of | #72304050, #72304082, #72304104, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1205549);