Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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:
(2,4)(13,14,17,20)(15,21,23,18)(16,24,25,19)(22,26,28,27)(29,30,33,36)(31,37,39,34)(32,35,42,41)(38,44,40,43);(6,9)(10,12)(16,25)(19,24)(22,28)(26,27)(29,31,33,39)(30,34,36,37)(32,40,42,38)(35,44,41,43);(5,6)(7,10)(8,9)(11,12)(13,15,17,23)(14,18,20,21)(16,22,25,28)(19,26,24,27)(29,31)(30,34)(32,38)(33,39)(35,43)(36,37)(40,42)(41,44);(1,2,3,4)(13,16,17,25)(14,19,20,24)(15,22,23,28)(18,26,21,27)(29,30)(31,37)(32,41)(33,36)(34,39)(35,42)(38,43)(40,44);(5,7,8,11)(6,10,9,12)(13,14)(15,18)(16,19)(17,20)(21,23)(22,26)(24,25)(27,28)(29,32,33,42)(30,35,36,41)(31,38,39,40)(34,43,37,44)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fg,d^2=fi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3046647);