Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #988785 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×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 | |
4052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(10,13)(11,15)(17,18)(19,23)(20,21)(22,24);(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,24)(18,22,20,23);(5,6,7,8)(11,15)(14,16)(18,20)(22,23);(1,2)(3,4)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,23)(21,24);(9,11,12,15)(10,14,13,16)(17,20,21,18)(19,23,24,22)Polycyclic presentation:
2^9|b^2=g,c^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72297338, #72297354, #72297407, … |
| maximal normal subgroup of | #72297338, #72297354, #72297407, … |
| maximal quotient of | #72298118, #72298200, #72298226, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 898209);