Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139432 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
671 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 671 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,10,7,11,26,30)(3,13,15,31,16,20,28,32)(4,18,14,23,21,9,25,17)(5,22,24,29,8,12,27,19);(1,3,14,11,5,23,32,12)(2,8,17,31,22,7,16,25)(4,19,24,28,20,30,6,18)(9,21,29,27,15,13,10,26);(4,20)(6,24)(9,15)(10,29)(13,21)(18,28)(19,30)(26,27)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=gh,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fh,g^b=gi,h^b=hi,d^c=dgh,f^c=fgh,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72237395, #72238434, #72238435, … |
| maximal normal subgroup of | #72237395, #72238434, #72238435, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46632);