Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1946 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)(7,8,11,10)(13,14,17,18)(15,16,19,20)(21,22,25,24)(23,26,27,28);(5,7,9,11)(6,10,12,8)(13,15,17,19)(14,16,18,20)(23,27)(26,28);(5,8,9,10)(6,11,12,7)(13,16)(14,15)(17,20)(18,19)(21,23)(22,26)(24,28)(25,27);(1,2,3,4)(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,19)(18,20);(5,7)(6,8)(9,11)(10,12)(13,14)(15,16)(17,18)(19,20)(21,24,25,22)(23,28,27,26)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,c^2=f,d^2=fi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1587824);