Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124996 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1543 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8344 |
| prime radical | #60693 | C2 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
610 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 610 conjugacy classes of subgroups — too many to draw. Only its 100 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:
(1,2,5,10,4,9,17,26,6,11,20,28,18,8,24,25)(3,12,14,29,15,23,31,32,16,21,30,27,13,7,19,22);(1,3)(2,7)(4,13)(5,19)(6,16)(8,12)(9,21)(10,27)(11,23)(14,24)(15,18)(17,30)(20,31)(22,25)(26,32)(28,29);(2,8,11,9)(5,20)(7,12,23,21)(10,26,28,25)(14,30)(17,24)(19,31)(22,27,32,29)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=h,e^2=i,f^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,f^b=fghi,g^b=gi,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72265723, #72265725 |
| maximal normal subgroup of | #72265723, #72265725 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37237);