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 |
| prime radical | #15412 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3054 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)(11,16)(12,13)(14,15)(19,20)(22,23);(6,8)(9,10,14,15)(11,16,12,13)(17,18,21,24)(19,22,20,23);(5,6)(7,8)(9,11,14,12)(10,13,15,16)(17,19,21,20)(18,22,24,23);(1,2)(3,4)(9,12,14,11)(10,13,15,16)(17,18)(19,22)(20,23)(21,24);(9,13)(10,11)(12,15)(14,16)(17,20,21,19)(18,23,24,22)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,d^2=f,e^2=fg,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 | #72281097, #72281245, #72282240, … |
| maximal normal subgroup of | #72281097, #72281245, #72282240, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2016440);