Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130512 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| prime radical | C22×D4 | C22⋊C4 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #2351 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
513 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 513 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(1,2,5,8,6,3,4,7)(9,10)(11,20)(12,25,27,17)(13,28,29,18)(14,19)(15,22,32,21)(16,26)(23,30,39,33)(24,36)(31,38)(34,40)(35,37);(1,3,6,2)(4,7,5,8)(9,11)(10,15)(12,22)(13,23)(14,24)(16,33)(17,20)(18,31)(19,32)(21,27)(25,36)(26,38)(28,35)(29,39)(30,34)(37,40);(9,12)(10,16,19,34)(11,21)(13,26)(14,27)(15,30,32,33)(17,18,25,28)(20,35,36,31)(22,24)(23,37)(29,40)(38,39)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=g,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,f^b=fgh,f^c=fgi,f^d=fi,f^e=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38150);