Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4307937 |
| automorphism | Aut(G) order 67108864 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(7,11)(10,12)(13,14,17,15)(16,19,20,18)(23,28)(24,26);(1,2,4,3)(6,9)(10,12)(14,15)(18,19)(21,22,25,27)(23,26,28,24);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,19,20,18)(21,23)(22,26)(24,27)(25,28);(21,24,25,26)(22,23,27,28);(5,7,8,11)(6,10,9,12)(13,16,17,20)(14,18,15,19)Polycyclic presentation:
2^9|a^2=fgi,b^2=g,c^2=f,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4215702);