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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,16)(17,21)(18,19)(20,24)(22,23)(25,27)(26,28)(29,30,33,34)(31,35,39,38)(32,40,41,36)(37,43,44,42);(5,7,9,12)(6,10,8,11)(14,19)(15,20)(17,25)(23,28)(30,34)(35,38)(36,40)(42,43);(6,8)(10,11)(13,15,18,20)(14,16,19,24)(17,26,25,22)(21,28,27,23)(29,31,33,39)(30,35,34,38)(32,37,41,44)(36,42,40,43);(1,2,3,4)(5,8,9,6)(7,11,12,10)(13,16)(14,20)(15,19)(17,23)(18,24)(21,26)(22,27)(25,28)(29,32)(30,36)(31,37)(33,41)(34,40)(35,42)(38,43)(39,44);(5,7)(6,10)(8,11)(9,12)(13,17,18,25)(14,21,19,27)(15,22,20,26)(16,23,24,28)(29,30)(31,38)(32,36)(33,34)(35,39)(37,43)(40,41)(42,44)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=f,d^2=fi,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, 5416960);