Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
5115 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)(5,6,7,8)(10,13)(14,16)(18,21)(19,23);(1,2,4,3)(9,10,12,13)(11,14,15,16)(17,18)(19,22)(20,21)(23,24);(2,3)(11,15)(14,16)(17,19,20,23)(18,22,21,24);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,24);(1,3,4,2)(9,11,12,15)(10,14,13,16)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^9|a^2=fhi,b^2=fgh,c^2=gh,e^2=fgh,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 | #72230894, #72232329, #72232340, … |
| maximal normal subgroup of | #72230894, #72232329, #72232340, … |
| maximal quotient of | #72264980, #72264988, #72278117, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2038716);