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 | #34655 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3004 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)(9,10,12,16,13,14,20,21)(11,17,18,24,19,15,22,23);(1,2)(3,4);(6,8);(2,4)(10,14)(15,17)(16,21)(23,24);(9,11,12,18,13,19,20,22)(10,15,16,23,14,17,21,24)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,d^b=df140 conjugacy classes — view the full character table.
| maximal subgroup of | #72263857, #72263860, #72263863, … |
| maximal normal subgroup of | #72263857, #72263860, #72263863, … |
| maximal quotient of | #72318326, #72319475, #72319491, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291216);