Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1776 |
| prime radical | C23 | #440 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
237 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 237 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,7,4,6,8,3)(9,10,12,17,13,19,23,16,14,18,11,20,22,24,21,15);(2,6)(3,7)(10,15,18,16)(11,21)(12,23)(13,22)(17,19,20,24);(1,3,8,6,4,7,5,2)(9,11,22,23,14,12,13,21)(10,16,19,20,18,15,24,17)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=g,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cg,d^a=di,f^a=fi,c^b=ch,d^b=dg,e^b=eg,g^b=gi,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225334, #72227173, #72227177, … |
| maximal normal subgroup of | #72225334, #72227173, #72227177, … |
| maximal quotient of | #72225332, #72258173 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36370);