Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137665 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1895 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9078 |
| prime radical | #3067 | C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
413 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 413 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(1,2,6,9,7,13,21,25,8,14,4,20,5,12,22,11)(3,15,19,31,18,30,32,24,16,29,17,27,23,10,26,28);(2,9,12,11,14,20,13,25)(3,16)(4,6)(10,24,29,31,30,28,15,27)(18,23)(21,22);(1,3)(2,10)(4,19)(5,23)(6,17)(7,18)(8,16)(9,24)(11,31)(12,29)(13,15)(14,30)(20,28)(21,26)(22,32)(25,27)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=cgh,d^b=di,f^b=fi,d^c=di,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214259, #72214645, #72214690, … |
| maximal normal subgroup of | #72214259, #72214645, #72214690, … |
| maximal quotient of | C2×#137665 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44861);