Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
2006 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)(6,10)(8,12)(15,19)(18,20)(21,22,27,32)(23,29,24,28)(25,34,35,30)(26,33,36,31);(6,10)(7,11)(14,17)(18,20)(21,23,27,24)(22,28,32,29)(25,31,35,33)(26,34,36,30);(5,6,9,10)(7,8,11,12)(13,14)(15,18)(16,17)(19,20)(21,24,27,23)(22,29,32,28)(25,31,35,33)(26,34,36,30);(1,2,3,4)(5,7,9,11)(6,8,10,12)(21,25)(22,30)(23,33)(24,31)(26,29)(27,35)(28,36)(32,34);(5,8)(6,11)(7,10)(9,12)(13,15)(14,18)(16,19)(17,20)(21,26)(22,31)(23,34)(24,30)(25,29)(27,36)(28,35)(32,33)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4571823);