Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
3896 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)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,22)(23,24);(6,8)(9,11,13,16)(10,14,15,12)(17,18,21,22)(19,24,23,20);(5,6,7,8)(11,16)(12,14)(18,22)(19,23);(1,2)(3,4)(9,12)(10,11)(13,14)(15,16)(17,19,21,23)(18,20,22,24);(9,10)(11,12)(13,15)(14,16)(17,20)(18,23)(19,22)(21,24)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fh,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
| maximal subgroup of | #72336309, #72336389, #72336449 |
| maximal normal subgroup of | #72336309, #72336389, #72336449 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1682457);