Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437418 |
| 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 | C2×#3488 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,9)(8,10)(11,12)(13,14)(15,23)(16,18)(17,21)(19,22)(20,24)(25,27)(26,28);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15,19,25)(14,20,22,28)(16,23,21,27)(17,24,18,26);(2,4)(6,10)(9,12)(13,16,19,21)(14,17,22,18)(15,23,25,27)(20,24,28,26);(2,4)(13,14)(15,20)(16,17)(18,21)(19,22)(23,24)(25,28)(26,27);(13,17,19,18)(14,21,22,16)(15,24,25,26)(20,27,28,23)Polycyclic presentation:
2^9|b^2=g,c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347538);