Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137582 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9064 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
409 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 409 conjugacy classes of subgroups — too many to draw. Only its 88 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:
(1,2,7,13,6,16,5,11,9,12,24,30,8,3,14,17)(4,18,21,32,20,29,25,27,23,10,26,28,22,15,19,31);(1,3,8,2,9,16,6,12)(4,15,20,10,23,29,22,18)(5,17,7,13,14,11,24,30)(19,27,21,32,25,31,26,28);(1,4)(2,10)(3,15)(5,21)(6,22)(7,25)(8,20)(9,23)(11,28)(12,18)(13,31)(14,26)(16,29)(17,32)(19,24)(27,30)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,d^2=g,e^2=ghi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,d^b=dg,e^b=egi,f^b=fgi,g^b=gi,d^c=dgi,e^c=eghi,f^c=fgi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72214114, #72214690, #72214756, … |
| maximal normal subgroup of | #72214114, #72214690, #72214756, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44780);