Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×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 | |
7603 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)(9,10,14,16)(11,13,12,15)(18,21)(19,23);(9,11,14,12)(10,15,16,13)(17,18,20,21)(19,22,23,24);(1,2)(3,4)(5,6)(7,8)(17,18)(19,22)(20,21)(23,24);(5,6)(7,8)(9,12,14,11)(10,13,16,15)(17,19,20,23)(18,22,21,24);(1,2)(3,4)(5,6)(7,8)(9,13)(10,12)(11,16)(14,15)(17,18)(19,24)(20,21)(22,23)Polycyclic presentation:
2^9|a^2=fh,b^2=f,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
| maximal subgroup of | #72230859, #72231349, #72232337, … |
| maximal normal subgroup of | #72230859, #72231349, #72232337, … |
| maximal quotient of | #72250297, #72278758, #72293945, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1326546);