Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #128836 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1762 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #442 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
286 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 286 conjugacy classes of subgroups — too many to draw. Only its 96 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:
(1,2,5,11,7,12,6,10)(3,13,14,16,15,4,9,8)(17,18,22,26,23,28,30,27)(19,21,29,32,25,20,24,31);(1,3,6,9,7,15,5,14)(2,8,10,4,12,16,11,13)(17,19,21,27,22,24,32,28,23,25,20,26,30,29,31,18);(1,4,5,8,7,13,6,16)(2,9,11,3,12,14,10,15)(17,20,30,32,23,21,22,31)(18,24,27,25,28,29,26,19)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ei,d^2=g,e^2=h,f^2=g,g^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,c^b=cgh,d^b=dh,e^b=ei,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285417, #72285438, #72285630, … |
| maximal normal subgroup of | #72285417, #72285438, #72285630, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36226);