Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,13,16)(11,14,18,23)(12,19,20,15)(17,24,21,22);(11,17)(12,20)(14,22)(15,19)(18,21)(23,24);(2,4)(9,11)(10,14)(12,21)(13,18)(15,24)(16,23)(17,20)(19,22);(6,8)Polycyclic presentation:
2^9|a^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,g^b=gi,e^c=ei,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72236668, #72236700, #72238108, … |
| maximal normal subgroup of | #72236668, #72236700, #72238108, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301269);