Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | |
1612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(7,11)(10,12)(13,14,17,20)(15,18,21,25)(16,23,24,19)(22,28,27,26)(29,30,33,34)(31,37,39,35)(32,40,41,36)(38,42,44,43);(2,4)(5,6,9,8)(7,10,11,12)(15,21)(18,25)(22,27)(26,28)(30,34)(35,37)(36,40)(42,43);(1,2)(3,4)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,24)(27,28)(29,31,33,39)(30,35,34,37)(32,38,41,44)(36,42,40,43);(5,7)(6,10)(8,12)(9,11)(13,15,17,21)(14,18,20,25)(16,22,24,27)(19,26,23,28)(29,30)(31,35)(32,40)(33,34)(36,41)(37,39)(38,43)(42,44);(5,8,9,6)(7,12,11,10)(13,16,17,24)(14,19,20,23)(15,22,21,27)(18,26,25,28)(29,32,33,41)(30,36,34,40)(31,38,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=f,d^2=fg,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2611207);