Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | |
4759 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)(9,10,13,16)(11,12,15,14)(17,18)(19,20)(21,22)(23,24);(6,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,23)(20,21)(22,24);(5,6,7,8)(11,15)(12,14)(18,21)(19,24);(1,2)(3,4)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23);(9,12)(10,15)(11,16)(13,14)(17,21,22,18)(19,23,24,20)Polycyclic presentation:
2^9|a^2=fg,c^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72278252, #72327133, #72327138, … |
| maximal normal subgroup of | #72278252, #72327133, #72327138, … |
| maximal quotient of | #72278481, #72278720, #72299190 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3944336);