Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #612424 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3086 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(6,8)(10,11)(14,17)(16,19)(21,22)(23,27)(24,25)(26,28);(1,2,4,3)(5,6,9,8)(7,10,12,11)(13,14,18,17)(15,19,20,16)(22,25)(26,27);(5,6,9,8)(7,10,12,11)(13,15,18,20)(14,19,17,16)(21,22)(23,26)(24,25)(27,28);(5,7,9,12)(6,10,8,11)(13,16)(14,15)(17,20)(18,19);(1,3,4,2)(5,8,9,6)(7,11,12,10)(13,17,18,14)(15,16,20,19)(21,23)(22,26)(24,28)(25,27)Polycyclic presentation:
2^9|b^2=fh,c^2=gi,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 523342);