Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| 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 | |
3992 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)(6,8)(9,10,14,13)(11,16,12,15)(17,18)(19,23)(20,22)(21,24);(9,11,14,12)(10,15,13,16)(17,18,20,22)(19,21,24,23);(1,2,4,3)(11,12)(15,16)(19,24)(21,23);(5,6)(7,8)(9,12,14,11)(10,15,13,16)(17,19,20,24)(18,21,22,23);(1,3,4,2)(9,13,14,10)(11,15,12,16)(17,18)(19,21)(20,22)(23,24)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232207, #72232212, #72232265, … |
| maximal normal subgroup of | #72232207, #72232212, #72232265, … |
| maximal quotient of | #72330977, #72331015, #72331583 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 999875);