Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,20)(16,19)(17,18)(22,23)(24,28);(6,7)(10,11)(13,15,18,16)(14,19,17,20)(22,23)(26,27);(8,12)(10,11)(14,17)(19,20)(21,22,25,23)(24,26,28,27);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,16,18,15)(14,19,17,20)(21,23,25,22)(24,27,28,26);(5,8,9,12)(6,10,7,11)(13,17,18,14)(15,20,16,19)(21,24,25,28)(22,26,23,27)Polycyclic presentation:
2^9|b^2=gh,c^2=h,d^2=fh,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3270721);