Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12108 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×#424 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
537 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 537 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 256 |
A minimal faithful permutation representation — 3 generators on 44 points:
(2,4)(5,6)(7,11,12,9)(8,10)(13,14,18,25,17,24,33,21,20,27,16,23,19,26,34,37)(15,28,31,36,35,22,38,40,32,42,29,43,30,41,44,39);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(14,21)(16,33)(17,19)(18,34)(22,36)(23,26)(24,25)(27,37)(28,40)(29,38)(30,35)(31,44)(39,42)(41,43);(5,7,8,12)(6,9,10,11)(13,15,20,32)(14,22,27,41)(16,31,18,29)(17,35,19,30)(21,36,37,43)(23,40,25,39)(24,42,26,28)(33,44,34,38)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=h,d^2=gi,e^2=i,f^2=gi,g^2=i,b^a=bd,c^a=ce,e^a=ehi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=ei,f^c=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44384);