Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1762 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
324 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 108 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:
(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(17,18,22,26,24,28,20,27)(19,21,29,32,25,23,30,31);(1,3)(2,6,8,15)(4,11)(5,10)(7,9,12,16)(13,14)(17,19,23,27,22,30,31,28,24,25,21,26,20,29,32,18);(3,10)(6,15)(9,16)(11,14)(17,20,24,22)(19,25)(21,32,23,31)(29,30)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=eg,d^2=g,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fg,g^a=gi,h^a=hi,c^b=cg,e^b=ei,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72287131, #72287179, #72287197, … |
| maximal normal subgroup of | #72287131, #72287179, #72287197, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35804);