Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2008 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)(5,6,8,10)(7,11,12,9)(15,19)(18,20)(21,22)(23,28)(24,29)(25,30)(26,27)(31,36)(32,33)(34,35);(5,6,8,10)(7,9,12,11)(14,17)(18,20)(22,27)(23,29)(30,35)(31,33);(5,6)(7,9)(8,10)(11,12)(13,14)(15,18)(16,17)(19,20)(21,23,26,29)(22,28,27,24)(25,33,34,31)(30,32,35,36);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(21,24,26,28)(22,29,27,23)(25,32,34,36)(30,33,35,31);(5,7)(6,9)(8,12)(10,11)(13,15,16,19)(14,18,17,20)(21,25,26,34)(22,30,27,35)(23,31,29,33)(24,32,28,36)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=f,d^2=f,e^2=gh,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, 4510024);