Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #128584 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C4×#102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8202 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×M32 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
303 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 303 conjugacy classes of subgroups — too many to draw. Only its 120 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:
(1,2,5,12,7,14,22,11,8,15,6,23,21,13,25,30)(3,16,4,10,17,31,19,28,18,27,20,9,24,26,32,29);(1,3,6,4,7,17,25,19,8,18,5,20,21,24,22,32)(2,9,11,27,13,28,12,31,15,10,30,16,14,29,23,26);(1,4,5,17,7,19,22,18,8,20,6,24,21,32,25,3)(2,10,12,31,14,28,11,27,15,9,23,26,13,29,30,16)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=g,e^2=h,f^2=g,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fgi,g^a=gi,c^b=ch,d^b=di,e^b=ei,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72214092, #72214153, #72214308, … |
| maximal normal subgroup of | #72214092, #72214153, #72214308, … |
| maximal quotient of | #72216234, C3×#128714, #72226299, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35910);