Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12324 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
426 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 44 points:
(2,4)(5,6,7,10,8,11,12,9)(13,14,17,23,19,26,36,42,20,27,37,24,18,25,35,41)(15,28,29,39,32,43,44,22,16,33,30,38,31,40,34,21);(1,2,3,4)(6,9)(7,12)(10,11)(13,15,20,16)(14,21,27,22)(17,34,37,44)(18,32,19,31)(23,40,24,43)(25,39,26,38)(28,42,33,41)(29,36,30,35);(13,16,20,15)(14,22,27,21)(17,34,37,44)(18,31,19,32)(23,40,24,43)(25,38,26,39)(28,42,33,41)(29,36,30,35)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=i,d^2=gi,e^2=h,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=ehi,f^c=fh,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44097);