Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
2051 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)(7,11)(8,9)(10,12)(14,17)(18,20)(21,22,26,29)(23,30,27,24)(25,28,34,36)(31,35,33,32);(5,6)(7,11)(8,9)(10,12)(15,19)(18,20)(21,23,26,27)(22,24,29,30)(25,31,34,33)(28,32,36,35);(6,9)(10,11)(13,14,16,17)(15,18,19,20)(21,23)(22,24)(25,33)(26,27)(28,35)(29,30)(31,34)(32,36);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,24)(22,27)(23,29)(25,32)(26,30)(28,33)(31,36)(34,35);(5,7,8,12)(6,10,9,11)(13,15)(14,18)(16,19)(17,20)(21,25,26,34)(22,28,29,36)(23,31,27,33)(24,32,30,35)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5387358);