Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(14,18)(15,19)(21,22,25,28)(23,29,31,26)(24,32,33,27)(30,34,36,35);(5,7,10,12)(6,11,8,9)(13,14,17,18)(15,20,19,16)(24,33)(27,32)(30,36)(34,35);(6,8)(9,11)(13,15,17,19)(14,16,18,20)(21,23,25,31)(22,26,28,29)(24,30,33,36)(27,34,32,35);(1,2)(3,4)(5,8,10,6)(7,9,12,11)(13,16)(14,15)(17,20)(18,19)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,33)(34,36);(5,9)(6,12)(7,8)(10,11)(13,14)(15,20)(16,19)(17,18)(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=f,c^2=fgh,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4049384);