Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #772778 |
| automorphism | ? |
| 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 | |
4506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(9,10)(11,12)(13,14)(15,16)(17,18)(19,24)(20,22)(21,23);(5,6,7,8)(10,14)(11,16)(17,19)(18,20)(21,22)(23,24);(1,2)(3,4)(5,6)(7,8)(9,11,13,16)(10,15,14,12)(17,20,21,24)(18,22,23,19);(9,12)(10,11)(13,15)(14,16)(17,19)(18,20)(21,22)(23,24);(9,10)(11,15)(12,16)(13,14)(17,18)(19,24)(20,22)(21,23)Polycyclic presentation:
2^9|a^2=fgi,b^2=fhi,c^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef98 conjugacy classes — view the full character table.
| maximal subgroup of | #72261498, #72261501, #72261504, … |
| maximal normal subgroup of | #72261498, #72261501, #72261504, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 704639);