Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #128808 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1762 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #417 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
300 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 110 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)(3,7)(4,6)(5,8)(9,10,14,18,16,19,15,22)(11,13,20,24,21,12,17,23);(2,6)(3,4)(7,8)(9,11,13,18,15,20,23,19,16,21,12,22,14,17,24,10);(1,3,5,4)(2,7,8,6)(9,12,14,23,16,13,15,24)(10,17,18,11,19,20,22,21)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=eh,d^2=g,e^2=i,f^2=gh,g^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,c^b=cg,d^b=dh,e^b=ei,d^c=di,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72309607, #72309612, #72309614, … |
| maximal normal subgroup of | #72309607, #72309612, #72309614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36124);