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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6)(7,12)(8,10)(9,11)(22,26)(24,31)(27,32)(30,35);(5,7,9,8)(6,10,11,12)(14,17)(18,20)(22,26)(23,29)(28,33)(30,35);(7,8)(10,12)(13,14,16,17)(15,18,19,20)(21,22,25,26)(23,27,29,32)(24,28,31,33)(30,34,35,36);(1,2)(3,4)(5,8,9,7)(6,10,11,12)(21,23)(22,27)(24,30)(25,29)(26,32)(28,34)(31,35)(33,36);(5,6)(7,12)(8,10)(9,11)(13,15,16,19)(14,18,17,20)(21,24)(22,28)(23,30)(25,31)(26,33)(27,34)(29,35)(32,36)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fh,d^2=f,e^2=fgh,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, 1682448);