Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9431 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1437 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
636 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 636 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9)(3,10)(4,6,11,8)(7,12)(13,16)(14,15)(17,18)(19,24,28,25)(20,30,29,32)(21,27)(22,31)(23,26);(2,6,9,8)(7,15,16,10)(17,19,22,29,23,28,21,20)(18,24,27,30,26,25,31,32);(1,3,11,13,5,14,4,12)(2,7,8,10,9,16,6,15)(17,20,21,28,23,29,22,19)(18,25,27,32,26,24,31,30)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=di,e^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72238860, C2×#110703, #72238922 |
| maximal normal subgroup of | #72238860, C2×#110703, #72238922 |
| maximal quotient of | C2×#110703, #72238899, #72238900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17899);