Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×#509 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(6,7)(10,15)(11,13)(12,19)(16,24)(17,23)(18,21)(20,22);(1,2,4,5)(3,7,8,6);(9,10)(11,16)(12,20)(13,22)(14,17)(15,19)(18,23)(21,24);(1,3,4,8)(2,6,5,7);(9,11,14,13)(10,16,17,22)(12,18,21,19)(15,20,23,24)Polycyclic presentation:
2^9|b^2=f,d^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,d^b=df,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72313401, #72313420, #72313496, … |
| maximal normal subgroup of | #72313401, #72313420, #72313496, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531076);