Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153295 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13422 |
| commutator, Frattini, cosocle | #532 | C23 |
| prime radical | #30496 | C2 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
401 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 401 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:
(2,8)(3,13)(4,6)(5,18)(9,24)(11,12)(14,29)(16,32)(20,21)(23,25)(27,30)(28,31);(1,2)(3,14)(4,11)(5,19)(6,8)(7,12)(9,16)(10,20)(13,27)(15,29)(17,30)(18,23)(21,25)(22,32)(24,28)(26,31);(1,3,4,15,7,13,6,17)(2,9,8,22,12,24,11,26)(5,16,18,31,20,28,21,32)(10,14,23,30,19,27,25,29)Polycyclic presentation:
2^9|c^2=d,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fgh,g^a=gi,c^b=cf,d^b=di,e^b=eg,f^b=fi,g^b=gi,h^b=hi,e^c=ehi,g^c=gh,f^e=fh,g^f=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72245391 |
| maximal normal subgroup of | #72245391 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60491);