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 | |
1784 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(6,8)(11,12)(13,14,17,16)(15,18,20,19)(21,22)(23,29)(24,26)(25,32)(27,31)(28,33)(30,34)(35,36);(5,6,10,8)(7,11,9,12)(13,14,17,16)(15,19,20,18)(22,24)(25,33)(27,29)(30,36);(5,7,10,9)(6,11,8,12)(13,15,17,20)(14,18,16,19)(21,23,26,31)(22,27,24,29)(25,30,33,36)(28,34,32,35);(1,2)(3,4)(5,8,10,6)(7,11,9,12)(21,24,26,22)(23,27,31,29)(25,32,33,28)(30,34,36,35);(5,9,10,7)(6,11,8,12)(13,16,17,14)(15,19,20,18)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,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, 1547448);