Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132549 |
| 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 | |
515 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 515 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,6,7)(9,10,15,21,16,22,12,28)(11,23,24,34,27,33,31,38)(13,18,14,20,29,19,30,37)(17,32,35,40,36,26,39,25);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17,22,36)(12,24)(13,25)(14,26)(15,31)(16,27)(18,33,19,23)(20,38,37,34)(21,35,28,39)(29,40)(30,32);(3,8)(5,7)(9,12,16,15)(10,18,22,19)(11,24,27,31)(13,30,29,14)(17,33,36,23)(20,21,37,28)(25,32,40,26)(34,39,38,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,d^2=hi,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=egh,f^b=fghi,d^c=dghi,e^c=ei,f^c=fg,e^d=ei,f^d=fi,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39934);