Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6,7,8)(10,14)(15,16)(17,18)(19,22)(20,24)(21,23);(1,2)(3,4)(9,10,13,14)(11,15,12,16)(17,19,22,18)(20,24,21,23);(2,4)(9,11,13,12)(10,15,14,16)(17,20,22,21)(18,23,19,24);(6,8)(9,11,13,12)(10,16,14,15)(17,21,22,20)(18,23,19,24);(9,12,13,11)(10,16,14,15)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=df116 conjugacy classes — view the full character table.
| maximal subgroup of | #72261684, #72261692, #72261745, … |
| maximal normal subgroup of | #72261684, #72261692, #72261745, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 517123);