Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12324 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1879 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ #1451 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
331 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 331 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,7,8)(9,10,13,19,15,21,31,39,16,22,12,18,14,20,32,40)(11,23,25,34,27,38,29,17,28,33,24,35,26,37,30,36);(1,2,6,5,4,7,3,8)(9,11)(10,17)(12,29)(13,30)(14,27)(15,26)(16,28)(18,37)(19,38)(20,35)(21,34)(22,36)(23,39)(24,31)(25,32)(33,40);(9,11)(10,17)(12,30)(13,29)(14,27)(15,26)(16,28)(18,38)(19,37)(20,35)(21,34)(22,36)(23,40)(24,32)(25,31)(33,39)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=gi,e^2=h,f^2=h,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, 44215);