Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1566 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)(5,6,7,8)(9,10,12,11)(14,19)(15,23)(16,18)(20,24)(25,26)(27,28);(2,4)(9,11,12,10)(13,14)(15,24)(16,21)(17,19)(18,22)(20,25)(23,27)(26,28);(6,8)(13,15,16,25,17,26,18,23)(14,20,22,24,19,27,21,28);(2,4)(13,16,17,18)(14,21,19,22)(15,25,26,23)(20,28,27,24)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=g,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72264366, #72264377, #72264505, … |
| maximal normal subgroup of | #72264366, #72264377, #72264505, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 277287);