Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #616960 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4286 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)(11,15)(12,16)(13,14)(19,23)(22,24);(10,14)(11,16)(18,21)(22,24);(5,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,18,20,21)(19,22,23,24);(1,2,3,4)(9,11)(10,12)(13,16)(14,15)(17,19)(18,22)(20,23)(21,24);(9,12)(10,11)(13,15)(14,16)Polycyclic presentation:
2^9|c^2=fh,d^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 | #72328091, #72328157, #72328207, … |
| maximal normal subgroup of | #72328091, #72328157, #72328207, … |
| maximal quotient of | #72329946, #72332551 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 533719);