Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13343 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C22×D4 | C22×C4 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | #1867 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
712 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 712 conjugacy classes of subgroups — too many to draw. Only its 138 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:
(2,4)(5,6,10,14,11,15,9,24)(7,16,20,27,21,35,19,34)(8,13,22,31,23,32,25,30)(12,26,28,36,29,17,33,18);(5,7,8,18)(6,12,13,16)(9,19,25,17)(10,20,22,26)(11,21,23,36)(14,28,31,27)(15,29,32,35)(24,33,30,34);(1,2,3,4)(5,8)(6,13)(7,17)(9,25)(10,22)(11,23)(12,27)(14,31)(15,32)(16,33)(18,20)(19,36)(21,26)(24,30)(28,35)(29,34)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=e,f^2=i,g^2=i,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,f^b=fhi,h^b=hi,d^c=dh,f^c=fi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72250465, #72257231, #72257245, … |
| maximal normal subgroup of | #72250465, #72257231, #72257245, … |
| maximal quotient of | #72265258, #72317580 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56477);