Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1714 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)(7,11)(8,9)(10,12)(13,14,17,18)(15,21,23,19)(16,24,25,20)(22,27,26,28)(31,37)(35,39)(38,43)(42,44);(6,9)(7,12)(14,18)(16,25)(19,21)(22,26)(29,30,34,32)(31,35,37,39)(33,40,41,36)(38,44,43,42);(5,6,8,9)(7,10,12,11)(13,15)(14,19)(16,26)(17,23)(18,21)(20,28)(22,25)(24,27)(29,31)(30,35)(32,39)(33,38)(34,37)(36,42)(40,44)(41,43);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,14)(15,19)(16,20)(17,18)(21,23)(22,27)(24,25)(26,28)(29,32,34,30)(31,35,37,39)(33,40,41,36)(38,42,43,44);(5,6)(7,10)(8,9)(11,12)(13,16,17,25)(14,20,18,24)(15,22,23,26)(19,27,21,28)(29,33,34,41)(30,36,32,40)(31,38,37,43)(35,42,39,44)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,c^2=h,d^2=fg,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5130774);