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 | |
1608 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)(15,21)(16,18)(17,20)(19,23)(22,27)(24,25)(26,28)(29,30,34,37)(31,38,32,35)(33,41,42,36)(39,43,40,44);(6,10)(11,12)(14,16)(17,25)(19,21)(22,28)(29,31,34,32)(30,35,37,38)(33,40,42,39)(36,44,41,43);(5,6,9,10)(7,11,8,12)(13,15,18,23)(14,19,16,21)(17,22,25,28)(20,26,24,27)(29,32,34,31)(30,35,37,38)(33,40,42,39)(36,43,41,44);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,16,18,14)(15,19,23,21)(17,20,25,24)(22,27,28,26)(29,30)(31,35)(32,38)(33,41)(34,37)(36,42)(39,44)(40,43);(5,8,9,7)(6,12,10,11)(13,17,18,25)(14,20,16,24)(15,22,23,28)(19,26,21,27)(29,33)(30,36)(31,39)(32,40)(34,42)(35,43)(37,41)(38,44)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fh,d^2=gh,e^2=gh,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, 2991696);