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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1872 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)(5,6,9,11)(7,8,12,10)(15,19)(18,20)(21,22,26,27)(23,28,30,24)(25,29,35,36)(31,34,33,32);(5,7,9,12)(6,10,11,8)(14,17)(18,20)(22,27)(23,30)(29,36)(31,33);(6,11)(8,10)(13,14,16,17)(15,18,19,20)(21,23)(22,28)(24,27)(25,33)(26,30)(29,32)(31,35)(34,36);(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,15,16,19)(14,18,17,20)(21,24,26,28)(22,23,27,30)(25,34,35,32)(29,33,36,31);(5,8,9,10)(6,7,11,12)(21,25,26,35)(22,29,27,36)(23,31,30,33)(24,32,28,34)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fgh,d^2=h,e^2=f,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, 4049422);