Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #108535 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C4⋊C4 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1770 |
| prime radical | C23 | #436 |
| LCS | G ▷ C42 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
239 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 239 conjugacy classes of subgroups — too many to draw. Only its 118 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:
(1,2,4,6,5,8,3,7)(9,10,13,18,12,17,21,24,14,19,22,11,16,15,23,20);(2,6)(3,4)(7,8)(10,15,19,17)(11,20,18,24)(13,22)(21,23);(1,3,5,4)(2,6,8,7)(9,11,21,10,16,24,13,15,14,18,23,19,12,20,22,17)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=deg,d^2=gi,e^2=h,h^2=i,b^a=bd,c^a=cg,f^a=fi,c^b=ch,d^b=dg,e^b=egi,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72309527, #72309529, #72309546, … |
| maximal normal subgroup of | #72309527, #72309529, #72309546, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36371);