Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D8 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C8×D8 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
261 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 261 conjugacy classes of subgroups — too many to draw. Only its 94 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 — 3 generators on 36 points:
(1,2)(3,4)(5,6,8,13,11,16,24,27,12,17,25,28,10,26,9,15)(7,18,20,31,23,34,36,32,19,29,35,33,22,30,21,14);(6,13,16,27,17,28,26,15)(7,19)(14,29,31,30,32,18,33,34)(20,35)(21,36)(22,23);(2,4)(5,7,8,20,11,23,24,36,12,19,25,35,10,22,9,21)(6,14,15,30,26,33,28,29,17,32,27,34,16,31,13,18)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=d,d^2=gi,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,c^b=ci104 conjugacy classes — view the full character table.
| maximal subgroup of | #72286922, #72320710, #72320718 |
| maximal normal subgroup of | #72286922, #72320710, #72320718 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47376);