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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1779 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)(7,9)(8,10)(11,12)(13,14,18,17)(15,19,16,20)(21,22,26,30)(23,29,25,27)(24,32,33,28)(31,35,34,36);(5,6,9,7)(8,10,12,11)(13,14)(15,20)(16,19)(17,18)(21,23,26,25)(22,27,30,29)(24,31,33,34)(28,35,32,36);(6,7)(8,12)(13,15,18,16)(14,19,17,20)(21,24)(22,28)(23,31)(25,34)(26,33)(27,35)(29,36)(30,32);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,16,18,15)(14,19,17,20)(21,25,26,23)(22,29,30,27)(24,31,33,34)(28,35,32,36);(5,8,9,12)(6,10,7,11)(13,17,18,14)(15,19,16,20)(21,22)(23,27)(24,32)(25,29)(26,30)(28,33)(31,36)(34,35)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=fg,d^2=i,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, 5417111);