Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129554 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#102 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
352 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 352 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(10,14,15,16)(17,18,20,25,21,26,30,23,22,27,19,24,29,32,28,31);(2,5)(6,7)(9,10,11,14,13,15,12,16)(18,23,27,31)(19,28)(20,30)(21,29)(24,26,25,32);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,14,15,16)(17,19,21,28,22,20,29,30)(18,24,26,31,27,25,32,23)Polycyclic presentation:
2^9|a^2=f,b^2=defgi,c^2=eg,d^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=ch,d^b=dg,f^b=fg,g^b=gi,h^b=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72343388, #72343390, #72343396, … |
| maximal normal subgroup of | #72343388, #72343390, #72343396, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49071);