Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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 | |
1674 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)(5,6,10,9)(7,12,11,8)(13,14,17,18)(15,21,23,19)(16,20,25,26)(22,28,24,27)(29,30)(31,37)(32,40)(33,36)(34,39)(35,41)(38,42)(43,44);(14,18)(19,21)(20,26)(27,28)(29,30,33,36)(31,37,39,34)(32,35,41,40)(38,43,44,42);(5,7)(6,8)(9,12)(10,11)(13,15,17,23)(14,19,18,21)(16,24,25,22)(20,28,26,27)(29,30)(31,34)(32,40)(33,36)(35,41)(37,39)(38,43)(42,44);(1,2,3,4)(5,8,10,12)(6,11,9,7)(13,14)(15,19)(16,20)(17,18)(21,23)(22,27)(24,28)(25,26)(29,31)(30,34)(32,38)(33,39)(35,42)(36,37)(40,43)(41,44);(5,9,10,6)(7,12,11,8)(13,16,17,25)(14,20,18,26)(15,22,23,24)(19,27,21,28)(29,32)(30,35)(31,38)(33,41)(34,42)(36,40)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fh,d^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4296720);