Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153308 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13424 |
| commutator, Frattini, cosocle | C2×#124 | C23 |
| prime radical | C2×#454 | C22 |
| LCS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
354 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 354 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)(3,13,17,8)(4,9)(5,19)(6,11)(7,10)(12,18)(14,23,15,30)(16,25,21,26)(20,27)(22,28)(24,31,29,32);(1,3)(2,8)(4,16)(5,14)(6,21)(7,17)(9,26)(10,13)(11,25)(12,29)(15,20)(18,32)(19,30)(22,31)(23,27)(24,28);(2,9,10,11)(3,14)(5,20)(8,23,13,30)(12,19,28,27)(15,17)(16,32)(18,22)(21,31)(24,25,29,26)Polycyclic presentation:
2^9|a^2=h,c^2=d,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ehi,f^a=fgh,g^a=gi,c^b=cf,e^b=eg,f^b=fi,g^b=gi,h^b=hi,e^c=eh,f^c=fi,g^c=gh,f^e=fh,g^f=gi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72219525, #72219527, #72219528, … |
| maximal normal subgroup of | #72219525, #72219527, #72219528, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60504);