Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13030 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#509 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
385 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 385 conjugacy classes of subgroups — too many to draw. Only its 104 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:
(1,2,4,7,5,3,6,8)(9,10,12,18,13,19,26,15,14,20,28,17,29,36,40,31)(11,21,22,33,24,37,27,34,25,32,23,38,30,35,39,16);(1,3,4,8,5,2,6,7)(10,15,20,31)(12,26)(13,29)(16,21,34,32)(17,19,18,36)(22,27)(23,39)(24,30)(28,40)(33,35,38,37);(9,11)(10,16)(12,27)(13,30)(14,25)(15,21)(17,35)(18,37)(19,38)(20,34)(22,26)(23,40)(24,29)(28,39)(31,32)(33,36)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=g,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,d^b=dg,e^b=eg,f^b=fg,g^b=gi,h^b=hi,d^c=dg,e^c=egi,f^c=fg,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216436, #72216443, #72216444, … |
| maximal normal subgroup of | #72216436, #72216443, #72216444, … |
| maximal quotient of | #72216590, C2×#136700 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43896);