Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #153457 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13561 |
| commutator | #127 | C22×C4 |
| Frattini, cosocle | #492 | C23 |
| prime radical | #3549 | C4 |
| LCS | G ▷ #127 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ #3152 ◁ G | |
| LECS | G ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
329 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 329 conjugacy classes of subgroups — too many to draw. Only its 46 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:
(1,2,4,9,6,11,17,25)(3,8,13,22,12,23,26,32)(5,7,15,20,18,10,16,24)(14,19,29,30,27,21,28,31);(2,7)(3,12)(8,19)(10,11)(13,26)(14,27)(20,24)(21,23)(28,29)(30,31);(1,3)(2,8)(4,13)(5,14)(6,12)(7,19)(9,22)(10,21)(11,23)(15,29)(16,28)(17,26)(18,27)(20,30)(24,31)(25,32)Polycyclic presentation:
2^9|a^2=e,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=ch,e^b=efhi,f^b=fhi,g^b=ghi,d^c=di,e^d=egh,f^d=fhi,g^d=gi,f^e=fhi,g^e=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72306642, #72306643, #72306645, … |
| maximal normal subgroup of | #72306642, #72306643, #72306645, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60659);