Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4414082 |
| automorphism | Aut(G) order 8388608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×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 | |
7532 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)(6,8)(10,14)(11,16)(19,23)(22,24);(9,10,13,14)(11,15,16,12)(18,21)(22,24);(5,6)(7,8)(10,14)(12,15)(17,18)(19,22)(20,21)(23,24);(1,2)(3,4)(5,6)(7,8)(9,11)(10,15)(12,14)(13,16);(9,12,13,15)(10,11,14,16)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^9|b^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72233687, #72233702, #72233711, … |
| maximal normal subgroup of | #72233687, #72233702, #72233711, … |
| maximal quotient of | #72330221, #72332509, #72333292, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4321285);