Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136885 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1874 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D4 | SD16 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
394 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 394 conjugacy classes of subgroups — too many to draw. Only its 100 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:
(1,2,4,6,5,8,3,7)(9,10,13,18,30,32,28,17,15,20,27,37,14,19,12,26)(11,21,23,16,31,33,22,38,25,39,40,35,24,36,29,34);(2,6)(3,4)(7,8)(9,11,14,31,15,25,30,24)(10,16,32,34,20,35,19,38)(12,23,27,22,28,40,13,29)(17,36,37,39,26,33,18,21);(2,6)(3,4)(7,8)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,d^2=g,e^2=h,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=eh,f^c=fh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72309442, #72309457, #72309460, … |
| maximal normal subgroup of | #72309442, #72309457, #72309460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44185);