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 | |
1924 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)(8,12)(9,10)(13,14,16,18)(15,17,20,19)(21,22)(23,27)(24,32)(25,26)(28,31)(29,34)(30,36)(33,35);(6,9)(11,12)(13,14)(15,19)(16,18)(17,20)(21,23,26,31)(22,27,25,28)(24,30,34,33)(29,35,32,36);(5,7,10,8)(6,11,9,12)(13,15,16,20)(14,17,18,19)(21,23)(22,28)(24,33)(25,27)(26,31)(29,35)(30,34)(32,36);(1,2,3,4)(5,8,10,7)(6,11,9,12)(13,14)(15,17)(16,18)(19,20)(21,24)(22,29)(23,30)(25,32)(26,34)(27,35)(28,36)(31,33);(5,9,10,6)(7,11,8,12)(13,15)(14,17)(16,20)(18,19)(21,25,26,22)(23,27,31,28)(24,32,34,29)(30,35,33,36)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=f,d^2=hi,e^2=fh,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, 5418055);