Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | |
4909 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)(10,14)(11,15)(17,18,20,22)(19,23,24,21);(5,6,7,8)(9,10,13,14)(11,12,15,16)(19,24)(21,23);(5,6)(7,8)(17,19)(18,21)(20,24)(22,23);(1,2)(3,4)(9,11)(10,12)(13,15)(14,16)(17,19)(18,21)(20,24)(22,23);(9,12)(10,15)(11,14)(13,16)(17,18)(19,21)(20,22)(23,24)Polycyclic presentation:
2^9|a^2=fi,b^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72234795, #72234823, #72234826, … |
| maximal normal subgroup of | #72234795, #72234823, #72234826, … |
| maximal quotient of | #72298318, #72323557, #72330158, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2514468);