Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×#102 |
| prime radical | C23×D8 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,7,9,8,10,11,12)(13,14)(15,18)(16,25)(17,21)(19,22)(20,23)(24,27)(26,28);(14,18)(15,22)(16,23)(19,20)(21,27)(24,28)(25,26);(2,4)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,d^b=dg,g^b=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72286666, #72323010, #72323077 |
| maximal normal subgroup of | #72286666, #72323010, #72323077 |
| maximal quotient of | #72286707 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36696);