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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(7,11)(9,12)(13,14,17,18)(15,21,22,19)(16,20,24,26)(23,28,25,27)(29,30)(31,36)(32,38)(33,41)(34,35)(37,42)(39,40)(43,44);(2,4)(5,6,8,10)(7,9,11,12)(14,18)(15,22)(16,24)(27,28)(30,35)(31,38)(33,42)(40,43);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,21)(16,20)(17,18)(19,22)(23,28)(24,26)(25,27)(29,31,34,38)(30,36,35,32)(33,39,42,44)(37,43,41,40);(5,7)(6,9)(8,11)(10,12)(13,15,17,22)(14,19,18,21)(16,25,24,23)(20,28,26,27)(29,32,34,36)(30,31,35,38)(33,43,42,40)(37,44,41,39);(13,16,17,24)(14,20,18,26)(15,23,22,25)(19,27,21,28)(29,33,34,42)(30,37,35,41)(31,39,38,44)(32,40,36,43)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=fg,d^2=f,e^2=f,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, 2993063);