Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2673 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3269 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)(7,11)(8,9)(10,12)(13,14)(15,16)(17,18)(19,20)(23,27)(26,28);(5,7)(6,10)(8,11)(9,12)(13,15,19,18)(14,17,20,16);(1,2)(3,4)(13,16,19,17)(14,15,20,18)(21,22,25,24)(23,26,27,28);(5,8,9,6)(7,11,12,10)(13,17,19,16)(14,15,20,18)(21,23)(22,26)(24,28)(25,27);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,18,19,15)(14,17,20,16)(21,24,25,22)(23,28,27,26)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fhi,d^2=f,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72256708, #72256742, #72256743, … |
| maximal normal subgroup of | #72256708, #72256742, #72256743, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1295241);