Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13657 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2177 |
| commutator | C16 | C2×C42 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C23 | #509 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
202 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 202 conjugacy classes of subgroups — too many to draw. Only its 106 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)(5,6,10,17,8,24,27,35,13,18,9,29,28,25,26,36,14,19,31,30,15,34,33,22,11,21,32,16,12,7,20,23);(5,7,11,24,14,25,13,34)(6,15,18,28,19,8,21,12)(9,30,10,16,32,17,31,29)(20,23,33,35,26,36,27,22);(1,3,4,2)(5,8,13,28,14,15,11,12)(6,7,21,34,19,25,18,24)(9,20,31,27,32,26,10,33)(16,35,30,23,29,22,17,36)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=deg,d^2=ghi,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ch,d^a=di,f^a=fi,d^b=dg,e^b=eg,g^b=gh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#13658, #72223644, C2×#152000, … |
| maximal normal subgroup of | C4×#13658, #72223644, C2×#152000, … |
| maximal quotient of | #72223657, #72223674, C2×#152000, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59196);