Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12948 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1864 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
308 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 308 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,3,4,6,8,7)(9,10,14,21,16,22,15,33)(11,23,25,37,27,34,26,38)(12,28,31,20,13,17,29,19)(18,30,35,24,36,32,39,40);(1,3,5,6,4,7,8,2)(10,17,22,28)(12,29)(13,31)(14,15)(18,34,36,23)(19,21,20,33)(24,30)(25,26)(32,40)(35,37,39,38);(9,11)(10,18)(12,30)(13,32)(14,26)(15,25)(16,27)(17,34)(19,37)(20,38)(21,39)(22,36)(23,28)(24,29)(31,40)(33,35)Polycyclic presentation:
2^9|a^2=f,b^2=fg,d^2=g,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dg,e^b=eg,f^b=fi,g^b=gi,d^c=dg,e^c=egi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216388, #72216443, #72216448, … |
| maximal normal subgroup of | #72216388, #72216443, #72216448, … |
| maximal quotient of | #72225152, C2×#135904 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43100);