Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134944 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1807 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#41 | SD16 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
273 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 273 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:
(1,2,4,7,5,3,8,6)(10,14,15,20)(11,17)(12,19)(16,23,22,21)(18,24);(9,10,11,14,12,16,17,21,13,15,18,20,19,22,24,23);(1,3)(2,5)(4,6)(7,8)(10,15)(14,20)(16,22)(21,23)Polycyclic presentation:
2^9|a^2=f,b^2=d,d^2=g,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ci,f^b=fi,e^c=eh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72287034, #72287115, #72287160, … |
| maximal normal subgroup of | #72287034, #72287115, #72287160, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42270);