Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #153436 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1402 |
| commutator | #127 | C22×C4 |
| Frattini, cosocle | #492 | C23 |
| prime radical | #3147 | C22 |
| LCS | G ▷ #127 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #3147 ◁ G | |
| LECS | G ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
311 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 48 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,11,7,14,4,9)(3,13,15,31,17,12,18,30)(5,8,20,25,22,27,32,29)(10,23,28,16,26,19,24,21);(1,3,7,17)(2,8,14,27)(4,18,6,15)(5,16,22,21)(9,28,11,24)(10,13,26,12)(19,32,23,20)(25,31,29,30);(1,4,7,6)(2,9,14,11)(3,15,17,18)(5,19,22,23)(8,24,27,28)(10,29,26,25)(12,30,13,31)(16,20,21,32)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=i,e^2=i,f^2=hi,g^2=i,b^a=bd,d^a=df,f^a=fg,g^a=gh,h^a=hi,c^b=ci,e^b=efhi,f^b=fhi,g^b=ghi,e^d=egh,f^d=fhi,g^d=gi,f^e=fhi,g^e=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72283541, #72283572, #72283655, … |
| maximal normal subgroup of | #72283541, #72283572, #72283655, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60640);