Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15315 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2659 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,12)(9,10)(13,14)(15,19)(16,17)(18,20)(23,24)(25,27);(5,7,9,12)(6,8,10,11)(14,16)(18,19)(21,22,26,28)(23,27,24,25);(6,10)(7,12)(13,15,17,20)(14,18,16,19)(21,23,26,24)(22,27,28,25);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,15)(14,18)(16,19)(17,20)(21,24,26,23)(22,25,28,27);(5,8)(6,7)(9,11)(10,12)(13,16,17,14)(15,19,20,18)(21,25)(22,23)(24,28)(26,27)Polycyclic presentation:
2^9|b^2=fgh,c^2=gh,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1004659);