Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D8 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#335 |
| 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)(5,6,8,16,10,17,26,34,12,19,27,14,11,18,9,15)(7,20,22,33,24,31,21,35,25,29,23,13,28,30,36,32);(5,7,9,23,11,24,27,36,12,25,26,22,10,28,8,21)(6,13,15,31,18,32,14,29,19,33,34,30,17,35,16,20);(1,3,4,2)(5,8,10,26,12,27,11,9)(6,14,17,15,19,16,18,34)(7,21,28,22,25,36,24,23)(13,29,35,31,33,20,32,30)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=deg,d^2=gi,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ci,d^b=dg,e^b=eg,g^b=gi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72216211, #72216225, #72216227, … |
| maximal normal subgroup of | #72216211, #72216225, #72216227, … |
| maximal quotient of | #72225375, #72225412 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36297);