Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #36434 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5453 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,18)(19,22)(20,21)(23,24);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(18,21)(22,24);(1,3,5,4)(2,7,8,6)(17,18)(19,22)(20,21)(23,24);(1,3,5,4)(2,7,8,6)(18,21)(22,24);(1,4,5,3)(2,6,8,7)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,c^b=ci,e^b=ei,d^c=di,e^d=ei140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537560);