Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
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 | |
4459 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,12,13)(11,15,16,14)(18,21)(19,23);(5,6,7,8)(10,13)(14,15)(19,23)(22,24);(5,6)(7,8)(10,13)(11,16)(17,18,20,21)(19,22,23,24);(1,2)(3,4)(9,11,12,16)(10,14,13,15)(17,18)(19,22)(20,21)(23,24);(9,10)(11,15)(12,13)(14,16)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fg,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232394, #72232397, #72232427, … |
| maximal normal subgroup of | #72232394, #72232397, #72232427, … |
| maximal quotient of | #72323544, #72330572, #72332784 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4040134);