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 | |
1683 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,9,7)(8,10,12,11)(13,14,18,19)(15,20,22,17)(16,24,26,21)(23,25,28,27)(29,30,35,32)(31,38,33,36)(34,37,42,40)(39,44,41,43);(8,12)(10,11)(14,19)(17,20)(21,24)(25,27)(29,31,35,33)(30,36,32,38)(34,41,42,39)(37,44,40,43);(6,7)(10,11)(13,15)(14,20)(16,23)(17,19)(18,22)(21,27)(24,25)(26,28)(29,32,35,30)(31,36,33,38)(34,37,42,40)(39,44,41,43);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16,18,26)(14,21,19,24)(15,23,22,28)(17,25,20,27)(29,33,35,31)(30,36,32,38)(34,41,42,39)(37,43,40,44);(5,8)(6,10)(7,11)(9,12)(13,17,18,20)(14,22,19,15)(16,25,26,27)(21,28,24,23)(29,34,35,42)(30,37,32,40)(31,39,33,41)(36,43,38,44)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=h,d^2=fg,e^2=f,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, 5382764);