Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153197 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13348 |
| commutator, Frattini, cosocle | #532 | C23 |
| prime radical | #33559 | 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 | |
524 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 524 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)(3,13,17,30)(4,8)(5,10)(6,21)(7,12)(9,15,29,19)(11,23)(14,27,32,28)(16,25,22,26)(18,31)(20,24);(2,8)(4,5)(9,25)(10,12)(11,24)(13,27)(15,19)(16,22)(18,20)(21,31)(26,29)(28,30);(1,3)(2,9)(4,15)(5,19)(6,22)(7,17)(8,13)(10,30)(11,27)(12,29)(14,18)(16,23)(20,32)(21,28)(24,25)(26,31)Polycyclic presentation:
2^9|a^2=h,d^2=hi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dhi,f^a=fghi,g^a=gi,c^b=cf,d^b=dhi,e^b=ei,g^b=gh,d^c=dg,e^c=ei,g^c=gi,h^c=hi,f^d=fh,g^f=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72279397 |
| maximal normal subgroup of | #72279397 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60393);