Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 2097152 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(6,9)(11,12)(15,19)(18,20)(23,26)(24,28);(5,6,10,9)(7,12,8,11)(14,17)(18,20)(21,22,25,27)(23,24,26,28);(5,7,10,8)(6,11,9,12)(13,14)(15,18)(16,17)(19,20)(21,23,25,26)(22,24,27,28);(1,2)(3,4)(5,8,10,7)(6,11,9,12)(21,24)(22,26)(23,27)(25,28);(5,9,10,6)(7,12,8,11)(13,15)(14,18)(16,19)(17,20)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=fg,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4327926);