Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| 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 | #34743 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2840 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)(5,6)(7,8)(9,10,14,12,13,17,23,22)(11,15,20,21,19,16,24,18);(1,2,3,4)(9,11)(10,15)(12,21)(13,19)(14,20)(16,17)(18,22)(23,24);(6,8)(9,11)(10,16)(12,18)(13,19)(14,20)(15,17)(21,22)(23,24);(9,12,14,17,13,22,23,10)(11,18,20,15,19,21,24,16);(11,19)(15,16)(18,21)(20,24)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,d^b=di,e^b=ei,e^c=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279972);