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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2138 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)(5,6,8,10)(7,11,12,9)(14,17)(15,19);(5,7,8,12)(6,9,10,11)(22,25)(26,28);(6,10)(9,11)(13,14)(15,18)(16,17)(19,20)(21,22,24,25)(23,26,27,28);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,16,19)(14,18,17,20);(5,7)(6,9)(8,12)(10,11)(13,14)(15,18)(16,17)(19,20)(21,23,24,27)(22,26,25,28)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=h,d^2=fg,e^2=h,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, 1002468);