Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994171 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #59112 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2050 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)(5,6,9,7)(8,10,12,11)(13,14)(15,18)(16,19)(17,20)(21,23)(22,26)(24,27)(25,28);(6,10)(7,11)(8,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28);(2,4)(13,15,17,23)(14,18,20,21)(16,24,25,22)(19,27,28,26);(5,7,9,6)(8,11,12,10)(13,16,17,25)(14,19,20,28)(15,22,23,24)(18,26,21,27);(14,20)(18,21)(19,28)(26,27)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,f^b=fi,d^c=dh,f^d=fi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72282003, #72282218, #72282786, … |
| maximal normal subgroup of | #72282003, #72282218, #72282786, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7031609);