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 | #3543 | C4 |
| LCS | G ▷ #127 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C2×#492 ◁ G | |
| LECS | G ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
337 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 337 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:
(1,2,4,9,6,11,18,27)(3,12,14,25,13,23,28,32)(5,8,16,24,19,10,17,26)(7,20,22,29,21,30,31,15);(1,3)(2,7)(4,14)(5,15)(6,13)(8,12)(9,25)(10,23)(11,21)(16,20)(17,30)(18,28)(19,29)(22,26)(24,31)(27,32);(3,13)(7,21)(12,23)(14,28)(15,29)(20,30)(22,31)(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=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 | #72306582, #72306592, #72306594, … |
| maximal normal subgroup of | #72306582, #72306592, #72306594, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60638);