Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #106 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#334 |
| prime radical | C23 | C2×#108 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
264 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6,8,14,10,16,25,15,12,18,26,33,11,17,9,27)(7,19,22,34,23,29,35,13,24,30,21,31,28,36,20,32);(5,7,9,22,11,23,26,35,12,24,25,21,10,28,8,20)(6,13,15,19,17,31,14,29,18,32,27,30,16,34,33,36);(1,3,4,2)(5,8,10,25,12,26,11,9)(6,14,16,15,18,33,17,27)(7,20,28,21,24,35,23,22)(13,29,34,19,32,36,31,30)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=g,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,f^a=fi,d^b=dg,e^b=egi,g^b=gi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72216215, #72216223, #72216225, … |
| maximal normal subgroup of | #72216215, #72216223, #72216225, … |
| maximal quotient of | C8×#1369, C2×#129090, #72227182 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36286);