Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×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 | |
5571 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)(6,8)(9,10,12,14)(11,13,16,15)(18,21)(22,23);(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,21)(22,24);(5,6,7,8)(17,19)(18,22)(20,24)(21,23);(1,2)(3,4)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,24)(21,22);(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,21)(23,24)Polycyclic presentation:
2^9|a^2=i,c^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232319, #72232364, #72232379, … |
| maximal normal subgroup of | #72232319, #72232364, #72232379, … |
| maximal quotient of | #72278224, #72279167, C2×#4420652, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4327848);