Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151326 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12186 |
| commutator, Frattini, cosocle | #407 | C23 |
| LCS | G ▷ #407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #407 ◁ G | |
| LECS | G ▷ #407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
408 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 36 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)(4,9)(5,18)(6,21)(7,11)(8,13)(10,20)(14,31)(15,29)(16,28)(17,26)(19,27)(22,23)(24,30)(25,32);(1,3)(2,8)(4,13)(5,19)(6,22)(7,15)(9,23)(10,27)(11,24)(12,18)(14,20)(16,32)(17,30)(21,25)(26,31)(28,29);(1,4,6,5,7,17,20,16)(2,9,21,18,11,26,10,28)(3,13,22,19,15,30,14,32)(8,23,25,12,24,31,27,29)Polycyclic presentation:
2^9|c^2=fgi,d^2=fi,f^2=i,b^a=bd,c^a=ce,d^a=df,f^a=fi,g^a=gh,d^b=df,e^b=egh,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=ehi,f^c=fi,g^c=gi,e^d=ehi,g^d=gi32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58523);