Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125945 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7754 |
| commutator, Frattini, cosocle | C2×#97 | C23 |
| prime radical | C2×#127 | D4 |
| LCS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
313 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 313 conjugacy classes of subgroups — too many to draw. Only its 54 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)(3,12,15,30)(4,17)(5,9)(6,11)(7,14,24,32)(8,21)(10,22)(13,26,16,29)(18,27)(19,23,31,25)(20,28);(4,18)(5,22)(8,17)(9,10)(12,26)(13,16)(14,31)(19,32)(20,21)(23,25)(27,28)(29,30);(1,3,6,15)(2,7)(4,19,20,31)(5,13,22,16)(8,26)(9,23)(10,25)(11,24)(12,17)(14,18,32,21)(27,29)(28,30)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=f,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,h^a=hi,d^b=df,e^b=egi,g^b=gi,h^b=hi,d^c=dg,e^c=eh,f^c=fi,g^c=gi,h^c=hi,e^d=ei38 conjugacy classes — view the full character table.
| maximal subgroup of | #72219580, #72226635, #72226636 |
| maximal normal subgroup of | #72219580, #72226635, #72226636 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33141);