Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134690 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#106 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(5,6,8,7)(9,10,15,19,14,18,16,24)(11,20,12,22,21,23,17,13);(1,2)(3,4)(5,7,8,6)(9,11,13,18,16,12,23,24,14,21,22,10,15,17,20,19);(2,4)(6,7)(9,12,13,24,16,21,23,10,14,17,22,19,15,11,20,18)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=h,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,g^b=gi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72309661, #72309669, #72309689, … |
| maximal normal subgroup of | #72309661, #72309669, #72309689, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56171);