Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #127645 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#340 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C25 | SD16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,11,15,13,17,19,18)(12,21,20,16,22,14,23,24);(1,3)(2,5,6,7)(4,8)(10,14,17,21)(11,19)(12,20)(15,16,18,24)(22,23);(3,8)(5,7)(9,11,13,19)(10,15,17,18)(12,20,22,23)(14,24,21,16)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=g,e^2=i,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,c^b=cgh,d^b=dh,e^b=ei,g^b=gi,h^b=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72279905, #72280015, #72280054, … |
| maximal normal subgroup of | #72279905, #72280015, #72280054, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34846);