Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #106 |
| commutator | C8 | C4×C16 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C22 | #1367 |
| prime radical | C23 | #364 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
154 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 64 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2)(3,4)(5,6,9,17,12,18,25,36,13,19,23,15,10,30,26,32,14,20,29,33,28,16,24,35,11,31,22,34,8,27,7,21);(2,4)(5,7,10,23,11,24,12,9,14,26,8,22,13,25,28,29)(6,15,18,32,19,33,30,35,20,34,16,21,31,17,27,36)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=f,f^2=i,g^2=h,h^2=i,b^a=bc,c^a=cf,e^a=ef,f^a=fi,d^b=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72216243, #72216264, #72216271, … |
| maximal normal subgroup of | #72216243, #72216264, #72216271, … |
| maximal quotient of | #72225409 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1215);