Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153131 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2199 |
| commutator, Frattini, cosocle | #532 | C23 |
| socle | C2 | #13348 |
| prime radical | #33564 | C2 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
493 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 493 conjugacy classes of subgroups — too many to draw. Only its 50 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:
(1,2,4,8)(3,12)(5,9,17,11)(6,21,18,27)(7,19)(10,24,22,20)(13,23)(14,26)(15,28)(16,25)(29,31)(30,32);(1,2)(3,13)(4,8)(5,9)(6,22)(7,15)(10,18)(11,17)(12,29)(14,30)(16,28)(19,31)(20,21)(23,26)(24,27)(25,32);(1,3)(2,7)(4,16)(5,19)(6,23)(8,26)(9,25)(10,28)(11,12)(13,22)(14,17)(15,18)(20,29)(21,31)(24,32)(27,30)Polycyclic presentation:
2^9|a^2=h,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dhi,f^a=fgh,g^a=gi,c^b=cf,d^b=dhi,f^b=fi,g^b=gh,d^c=dg,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fh,g^f=gi35 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60392);