Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1756 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)(6,9)(10,12)(13,14)(15,20)(16,22)(17,21)(18,19)(23,25)(24,28)(26,27)(29,30,34,37)(31,38,32,35)(33,41,42,36)(39,43,40,44);(7,11)(10,12)(14,19)(15,22)(21,27)(23,28)(29,31,34,32)(30,35,37,38)(33,39,42,40)(36,43,41,44);(5,6,8,9)(7,10,11,12)(13,15,18,22)(14,20,19,16)(17,23,26,28)(21,25,27,24)(29,32,34,31)(30,35,37,38)(33,40,42,39)(36,43,41,44);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,20)(14,15,19,22)(17,25,26,24)(21,28,27,23)(29,30)(31,35)(32,38)(33,41)(34,37)(36,42)(39,44)(40,43);(5,7)(6,10)(8,11)(9,12)(13,17,18,26)(14,21,19,27)(15,23,22,28)(16,24,20,25)(29,33,34,42)(30,36,37,41)(31,39,32,40)(35,43,38,44)Polycyclic presentation:
2^9|a^2=fh,b^2=fh,c^2=fg,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5524346);