Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2424 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)(7,11)(10,12)(13,14,17,18)(15,19,20,16)(22,23)(24,28);(6,9)(10,12)(14,18)(16,19)(21,22,25,23)(24,27,28,26);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,20)(14,19,18,16)(21,23,25,22)(24,26,28,27);(1,2)(3,4)(13,16,17,19)(14,20,18,15)(21,24,25,28)(22,26,23,27);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=fg,b^2=fgi,c^2=f,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4837265);