Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,7,8)(9,10)(11,13)(12,16)(14,15)(17,18)(19,23)(20,21)(22,24);(1,2)(3,4)(10,15)(11,12)(17,18,21,20)(19,22,24,23);(9,11,14,12)(10,16,15,13)(17,19)(18,22)(20,23)(21,24);(2,4)(9,12,14,11)(10,16,15,13)(17,19)(18,23)(20,22)(21,24);(1,2)(3,4)(9,13,14,16)(10,12,15,11)(17,20,21,18)(19,23,24,22)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=h,d^2=h,e^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^d=ef140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 426470);