Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
503 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 503 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9)(3,7,11,16)(4,6,12,8)(10,15,13,14)(17,18,22,19)(20,27,29,32)(21,26,31,24)(23,30,25,28);(2,6)(4,12)(7,15)(8,9)(10,13)(14,16)(17,19,22,18)(20,25,29,23)(21,24,31,26)(27,28,32,30);(1,3,5,11)(2,7,9,16)(4,13,12,10)(6,14,8,15)(17,20)(18,23)(19,25)(21,30)(22,29)(24,32)(26,27)(28,31)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=g,d^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fghi,c^b=cf,d^b=dg,e^b=eh,d^c=dg,e^c=ei,h^c=hi,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51533);