Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130498 |
| 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×#127 | C2×C4 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#557 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
670 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 670 conjugacy classes of subgroups — too many to draw. Only its 84 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:
(2,6,7,8)(9,10,14,18,15,20,12,26)(11,21,25,16,22,33,23,34)(13,17,27,36,29,19,30,35)(24,39,38,32,28,37,40,31);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16)(12,23)(13,28)(14,25)(15,22)(17,32)(18,21)(19,31)(20,34)(24,29)(26,33)(27,40)(30,38)(35,37)(36,39);(1,3,5,4)(2,6,7,8)(10,17)(11,22)(12,14)(13,29)(16,31)(18,36)(19,20)(21,37)(26,35)(32,34)(33,39)(38,40)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=ei,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, 37762);