Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144402 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10337 |
| commutator, Frattini, cosocle | C22×#41 | C23 |
| prime radical | #23896 | C2 |
| LCS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
733 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 733 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:
(2,7,10,14)(3,11)(8,20,23,32)(9,19,24,29)(12,26)(13,27)(18,25,21,22)(28,31);(1,2)(3,8)(4,14)(5,9)(6,10)(7,16)(11,25)(12,18)(13,23)(15,29)(17,24)(19,30)(20,31)(21,28)(22,27)(26,32);(1,3)(2,8)(4,12)(5,11)(6,13)(7,18)(9,22)(10,23)(14,21)(15,26)(16,28)(17,27)(19,20)(24,25)(29,32)(30,31)Polycyclic presentation:
2^9|a^2=g,d^2=i,b^a=bd,c^a=ce,f^a=fgh,c^b=cf,d^b=di,e^b=eh,g^b=gi,d^c=dg,h^c=hi,f^d=fi,f^e=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72338780 |
| maximal normal subgroup of | #72338780 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51598);