Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2250 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)(7,9)(8,12)(10,11)(14,19)(15,16)(21,22,26,28)(23,24,27,25);(6,11)(7,8)(13,14,18,19)(15,17,16,20)(21,23)(22,25)(24,28)(26,27);(5,7,10,8)(6,12,11,9)(13,15,18,16)(14,17,19,20)(21,24,26,25)(22,27,28,23);(1,2)(3,4)(5,8,10,7)(6,12,11,9)(13,16,18,15)(14,17,19,20)(21,23)(22,24)(25,28)(26,27);(5,9,10,12)(6,8,11,7)(13,17)(14,16)(15,19)(18,20)(21,25,26,24)(22,27,28,23)Polycyclic presentation:
2^9|a^2=ghi,b^2=fh,c^2=f,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4776323);