Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139424 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9428 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C4×M16 | C23 |
| prime radical | #532 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C4×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
342 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 60 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:
(1,2,7,13,6,27,26,11,8,12,31,32,29,14,5,23)(3,15,18,4,19,22,30,21,17,20,24,9,10,25,16,28);(1,3,6,10,8,17,29,19)(2,9,27,21,12,4,14,28)(5,24,31,16,26,18,7,30)(11,22,13,20,23,25,32,15);(1,4)(2,10)(3,14)(5,25)(6,28)(7,20)(8,9)(11,16)(12,19)(13,18)(15,31)(17,27)(21,29)(22,26)(23,30)(24,32)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,d^2=gh,e^2=gi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fhi,g^b=gi,d^c=dgh,e^c=egi,f^c=fgh,h^c=hi,f^d=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72306588, #72306763, #72306767, … |
| maximal normal subgroup of | #72306588, #72306763, #72306767, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46620);