Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573549 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3307 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(13,14,16,20,17,19,25,27)(15,21,22,26,23,28,24,18);(5,6,8,9)(7,10,11,12)(13,15,17,23)(14,18,19,26)(16,24,25,22)(20,28,27,21);(2,4)(6,9)(7,11);(5,7)(6,10)(8,11)(9,12)(14,19)(18,26)(20,27)(21,28);(6,9)(7,11)Polycyclic presentation:
2^9|a^2=f,b^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=eh83 conjugacy classes — view the full character table.
| maximal subgroup of | #72257442, #72257464, #72263859, … |
| maximal normal subgroup of | #72257442, #72257464, #72263859, … |
| maximal quotient of | #72305809, #72305925, #72306064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503280);