Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11875 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | #408 | C23 |
| prime radical | #28177 | C2 |
| LCS | G ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2932 ◁ G | |
| LECS | G ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
300 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 58 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:
(1,2,5,11)(3,14,17,8)(4,19,18,29)(6,13,24,28)(7,12,16,25)(9,22,10,30)(15,31,20,26)(21,27,23,32);(2,8)(3,15)(4,20)(5,23)(6,21)(9,25)(10,13)(11,29)(12,26)(14,28)(16,24)(17,22)(19,27)(31,32);(1,3,6,18,7,15,21,30)(2,9,27,31,12,29,13,14)(4,5,20,24,22,16,17,23)(8,25,10,28,26,11,19,32)Polycyclic presentation:
2^9|a^2=e,c^2=gh,d^2=gi,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=cf,d^b=dg,e^b=efg,g^b=gi,d^c=dhi,e^c=ei,f^c=fi,e^d=eh,f^e=fi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58008);