Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94188 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1390 |
| commutator | C2×C4⋊C4 | C42 |
| Frattini, cosocle | C4×#127 | C22 |
| prime radical | C2×#127 | D4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C4×#127 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
286 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 286 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,6)(3,4)(7,8)(9,10)(11,19,21,24)(12,18,13,16)(14,17)(15,20,23,22);(1,2,4,7,5,8,3,6)(9,11)(10,15,17,24)(12,22)(13,21)(14,20)(16,23,18,19)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fg,e^2=fg,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fh,g^a=gh,h^a=hi,c^b=ci,d^b=dg,f^b=fi,g^b=gi,h^b=hi,d^c=dhi,f^c=fi,g^c=gi,f^d=fi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1421);