Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #616960 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2613 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5177 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6,7,8)(9,10)(11,15)(12,13)(14,16)(17,18,20,21)(19,23,24,22);(2,4)(10,13)(14,15)(18,21)(22,23);(1,2,3,4)(5,6)(7,8)(17,18)(19,22)(20,21)(23,24);(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,24);(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)Polycyclic presentation:
2^9|a^2=g,c^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72300414, #72300426, #72300437, … |
| maximal normal subgroup of | #72300414, #72300426, #72300437, … |
| maximal quotient of | #72323499, #72329892, #72329896, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 529587);