Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140467 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9618 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #471 |
| 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 | |
201 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 201 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(2,9,11,22)(3,13,16,28)(4,19)(5,21)(6,7)(10,14)(12,26,23,20)(15,17,29,18)(24,32)(25,31);(1,2,6,23,8,11,7,12)(3,14,18,25,16,31,17,10)(4,20,19,9,21,26,5,22)(13,27,29,24,28,30,15,32);(1,3,5,13,7,18,21,29,8,16,19,28,6,17,4,15)(2,10,9,24,12,14,20,30,11,25,22,32,23,31,26,27)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=e,d^2=g,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ch,d^b=dg,e^b=ei,f^b=fi,g^b=gi,f^c=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72214066, #72214178, #72214308, … |
| maximal normal subgroup of | #72214066, #72214178, #72214308, … |
| maximal quotient of | #72216390, #72227276, #72227277, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47663);