Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153307 |
| 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×H(5,𝔽2) | D4 |
| 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 | |
356 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 356 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)(3,13,16,7)(4,8)(5,11)(6,9)(10,19)(12,21)(14,29,30,22)(15,23,20,28)(17,27,31,24)(18,25)(26,32);(1,3)(2,7)(4,17)(5,20)(6,16)(8,24)(9,13)(10,27)(11,23)(12,28)(14,18)(15,21)(19,31)(22,25)(26,29)(30,32);(2,8,9,10)(7,22,13,29)(11,25,12,26)(14,30)(15,20)(23,24,28,27)Polycyclic presentation:
2^9|a^2=h,c^2=d,e^2=hi,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,g^b=gi,h^b=hi,e^c=eh,g^c=gh,f^e=fh,g^f=gi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72219523, #72219527, #72219528, … |
| maximal normal subgroup of | #72219523, #72219527, #72219528, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60503);