Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142604 |
| 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 | |
509 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 509 conjugacy classes of subgroups — too many to draw. Only its 68 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:
(1,2,7,11)(3,12,4,15)(5,10,6,9)(8,14,16,13)(17,18,21,24)(19,22,27,32)(20,28,29,23)(25,30,26,31);(1,3,7,4)(2,8,11,16)(5,14,6,13)(9,15,10,12)(17,19)(18,22)(20,25)(21,27)(23,30)(24,32)(26,29)(28,31);(1,4,7,3)(2,8,11,16)(5,13,6,14)(9,15,10,12)(19,25)(20,29)(22,30)(23,28)(26,27)(31,32)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,d^2=g,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,f^b=fi,h^b=hi,d^c=dh,e^c=eg,f^c=fi,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49978);