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) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2801 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)(5,6,9,12)(7,8,11,10)(13,14,17,16)(15,19,20,18)(23,27)(26,28);(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,19)(17,20)(22,25)(26,28);(1,2,4,3)(13,15)(14,18)(16,19)(17,20)(21,22)(23,26)(24,25)(27,28);(1,3,4,2)(5,7)(6,8)(9,11)(10,12)(13,16,17,14)(15,19,20,18);(5,8)(6,11)(7,12)(9,10)(13,15)(14,19)(16,18)(17,20)(21,23)(22,26)(24,27)(25,28)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=g,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, 4337602);