Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2150 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)(13,14,18,21)(15,22,24,19)(16,25,17,20)(23,27,26,28);(5,6,8,11)(7,9,12,10)(13,15,18,24)(14,19,21,22)(16,23,17,26)(20,27,25,28);(2,4)(6,9)(7,12)(10,11);(13,16,18,17)(14,20,21,25)(15,23,24,26)(19,27,22,28);(6,10)(7,12)(9,11)(13,17,18,16)(14,20,21,25)(15,23,24,26)(19,28,22,27)Polycyclic presentation:
2^9|a^2=f,b^2=fi,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72283920, #72284243, #72285057, … |
| maximal normal subgroup of | #72283920, #72284243, #72285057, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6584428);