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) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34925 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2812 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)(9,10,14,18,13,15,23,20)(11,19,21,22,17,16,24,12);(1,2)(3,4)(10,15)(11,17)(18,20)(21,24);(6,8)(9,11)(10,16)(12,18)(13,17)(14,21)(15,19)(20,22)(23,24);(2,4)(10,15)(11,17)(18,20)(21,24);(9,12,14,19,13,22,23,16)(10,17,18,24,15,11,20,21)Polycyclic presentation:
2^9|a^2=h,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ci,d^b=dfi,d^c=di,e^c=ei140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6290983);