Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×Q8 | C22×C8 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| prime radical | C22×D4 | C22⋊C4 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
655 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 655 conjugacy classes of subgroups — too many to draw. Only its 112 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:
(1,2,3,5,4,6,7,8)(9,10)(11,21,24,35)(12,18,13,26)(14,19)(15,20)(16,22)(17,25,34,28)(23,37,40,30)(27,32)(29,33)(31,38)(36,39);(9,11,15,24)(10,16,20,32)(12,22,13,27)(14,23,29,40)(17,30,34,37)(18,35,26,21)(19,36,33,31)(25,38,28,39);(9,12)(10,17)(11,22)(13,15)(14,28)(16,30)(18,33)(19,26)(20,34)(21,36)(23,39)(24,27)(25,29)(31,35)(32,37)(38,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=di,e^b=eg,f^b=fgi,d^c=dgi,e^c=ei,f^c=fi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36782);