Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136906 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #33580 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
429 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 429 conjugacy classes of subgroups — too many to draw. Only its 90 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:
(1,2,6,7,5,3,4,8)(9,10,13,18,14,19,12,24,16,21,11,22,15,20,23,17);(1,3,5,2)(4,8,6,7)(10,17,21,24)(11,23)(12,13)(14,15)(18,19,22,20);(1,4,5,6)(2,7,3,8)(10,18)(11,12)(13,23)(14,15)(17,19)(20,24)(21,22)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=gh,d^2=g,e^2=h,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=ehi,f^c=fh,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72342413, #72342417, #72342422, … |
| maximal normal subgroup of | #72342413, #72342417, #72342422, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44135);