Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×#102 |
| prime radical | C2×#3169 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
796 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 796 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,8,11,9,7,12,10)(13,14)(15,18)(16,25)(17,21)(19,22)(20,23)(24,27)(26,28);(5,7)(6,9)(8,10)(11,12)(14,18)(15,22)(16,23)(19,20)(21,27)(24,28)(25,26);(2,4)(6,7)(10,11)Polycyclic presentation:
2^9|a^2=f,d^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=dg,g^b=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72321045, #72323010, #72323125 |
| maximal normal subgroup of | #72321045, #72323010, #72323125 |
| maximal quotient of | #72321070 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36747);