Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2031997 |
| automorphism | ? |
| 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 | |
2838 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)(5,6,9,12)(7,11,8,10)(13,14,16,18)(15,19,20,17)(22,24)(25,28);(5,7,9,8)(6,10,12,11)(13,14,16,18)(15,17,20,19)(21,22,26,24)(23,28,27,25);(5,7,9,8)(6,11,12,10)(13,15)(14,17)(16,20)(18,19)(21,23)(22,25)(24,28)(26,27);(1,2)(3,4)(5,8,9,7)(6,10,12,11)(13,14)(15,17)(16,18)(19,20)(21,24,26,22)(23,28,27,25);(5,6)(7,11)(8,10)(9,12)(13,15)(14,17)(16,20)(18,19)(21,25,26,28)(22,27,24,23)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=fh,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1946341);