Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153125 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2199 |
| commutator, Frattini, cosocle | #532 | C23 |
| socle | C2 | #13419 |
| prime radical | #31076 | 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 | |
423 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 423 conjugacy classes of subgroups — too many to draw. Only its 46 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)(3,15)(4,19)(5,18)(6,17)(7,12)(8,11)(9,14)(10,21)(13,24)(16,25)(20,27)(22,28)(23,30)(26,32)(29,31);(1,3)(2,6)(4,10)(5,12)(7,14)(8,17)(9,13)(11,24)(15,18)(16,22)(19,32)(20,31)(21,27)(23,28)(25,26)(29,30);(1,4,5,20,9,23,8,25)(2,10,11,26,14,28,18,31)(3,16,17,30,13,27,12,19)(6,21,15,29,7,22,24,32)Polycyclic presentation:
2^9|c^2=d,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fghi,h^a=hi,c^b=cf,d^b=di,e^b=eg,f^b=fi,g^b=gi,h^b=hi,g^c=gh,f^e=fhi,g^e=gi,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60477);