Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131128 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| prime radical | C22×D4 | C22⋊C4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
520 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 520 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:
(2,5,6,7)(9,10,14,19,15,20,12,24)(11,16,21,31,23,32,28,39)(13,18,25,33,27,35,26,17)(22,30,37,29,38,34,36,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,21)(13,22)(14,28)(15,23)(17,29)(18,34)(19,31)(20,32)(24,39)(25,36)(26,37)(27,38)(30,35)(33,40);(9,12,15,14)(10,17,20,33)(11,21,23,28)(13,26,27,25)(16,29,32,40)(18,19,35,24)(22,37,38,36)(30,39,34,31)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=egi,f^b=fgh,d^c=dg,e^c=ei,f^c=fgi,e^d=ei,f^d=fi,f^e=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38332);