Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 1048576 |
| 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) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,10)(8,12)(9,11)(13,14,18,16)(15,17,20,19)(21,22)(23,29)(24,26)(25,32)(27,31)(28,33)(30,35)(34,36);(5,7,9,12)(6,10,11,8)(15,20)(17,19)(22,24)(25,33)(27,29)(30,34);(5,8)(6,7)(9,10)(11,12)(13,15,18,20)(14,19,16,17)(21,23,26,31)(22,27,24,29)(25,34,33,30)(28,36,32,35);(1,2)(3,4)(5,8)(6,7)(9,10)(11,12)(13,16,18,14)(15,17,20,19)(21,24,26,22)(23,29,31,27)(25,28,33,32)(30,35,34,36);(5,7)(6,8)(9,12)(10,11)(13,17)(14,15)(16,20)(18,19)(21,25,26,33)(22,28,24,32)(23,30,31,34)(27,35,29,36)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=fh,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3951985);