Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132253 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| prime radical | C2×#562 | C4 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#557 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
645 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 645 conjugacy classes of subgroups — too many to draw. Only its 80 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:
(2,5,6,7)(9,10,14,19,15,20,12,24)(11,16,21,32,23,33,29,39)(13,18,25,34,28,36,27,17)(22,31,38,30,26,35,37,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,21)(13,26)(14,29)(15,23)(17,30)(18,35)(19,32)(20,33)(22,28)(24,39)(25,38)(27,37)(31,36)(34,40);(3,8)(5,7)(9,12,15,14)(10,17,20,34)(11,21,23,29)(13,27,28,25)(16,30,33,40)(18,19,36,24)(22,38,26,37)(31,39,35,32)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=ehi,f^b=fgh,d^c=dhi,e^c=ei,f^c=fgi,f^d=fi,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39454);