Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11663 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×#135 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,7,9,8,10,11,12)(14,19,21,28)(15,18)(16,23)(20,24,26,27)(22,25);(13,14)(15,20)(16,24)(17,21)(18,26)(19,22)(23,28)(25,27);(2,4)(13,15,17,18)(14,20,21,26)(16,22,25,23)(19,27,28,24)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dgi,f^b=fi,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72318742, #72318761, #72318764, … |
| maximal normal subgroup of | #72318742, #72318761, #72318764, … |
| maximal quotient of | #72322507, #72322636 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36701);