Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,8,11,9,7,10,12)(14,18,20,24)(15,21)(19,26,27,22)(23,28);(5,7,8,12,9,6,10,11)(13,14,16,24,17,20,25,18)(15,22,23,19,21,26,28,27);(2,4);(13,15,16,23,17,21,25,28)(14,19,18,22,20,27,24,26)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,d^b=de,e^b=eh,g^b=gh116 conjugacy classes — view the full character table.
| maximal subgroup of | #72318512, #72318555, #72318570, … |
| maximal normal subgroup of | #72318512, #72318555, #72318570, … |
| maximal quotient of | #72322476 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 136304);