Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1896 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)(7,8)(9,10)(11,12)(15,19)(18,20)(21,22,25,28)(23,29,31,26)(24,32,33,27)(30,34,36,35);(5,7)(6,11)(8,9)(10,12)(14,17)(18,20)(21,22,25,28)(23,29,31,26)(24,27,33,32)(30,35,36,34);(5,8,10,11)(6,12,9,7)(13,14,16,17)(15,18,19,20)(21,23,25,31)(22,26,28,29)(24,30,33,36)(27,34,32,35);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(13,15,16,19)(14,18,17,20)(21,22)(23,26)(24,27)(25,28)(29,31)(30,34)(32,33)(35,36);(5,9,10,6)(7,8,12,11)(21,24)(22,27)(23,30)(25,33)(26,34)(28,32)(29,35)(31,36)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=h,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3252572);