Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139706 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1889 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | #3484 | C22 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
742 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 742 conjugacy classes of subgroups — too many to draw. Only its 68 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)(3,15,18,27)(4,9,16,26)(5,14)(6,21,24,31)(7,13)(8,11)(10,19,29,17)(12,20,30,23)(22,28,32,25);(1,3,5,4)(2,9,11,27)(6,19,20,17)(7,18,8,16)(10,21,25,31)(12,29,30,28)(13,26,14,15)(22,23,32,24);(1,4,5,3)(2,10,14,29)(6,22,20,32)(7,16,8,18)(9,21,15,30)(11,25,13,28)(12,27,31,26)(17,24,19,23)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=gi,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72236605, #72236732, #72236741, … |
| maximal normal subgroup of | #72236605, #72236732, #72236741, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46964);