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 | |
1698 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)(10,12)(13,14,18,15)(16,21,24,19)(17,22,25,20)(23,27,26,28)(29,30)(31,34)(32,37)(33,36)(35,39)(38,41)(40,44)(42,43);(1,2,3,4)(14,15)(17,25)(19,21)(23,26)(30,31)(32,39)(33,41)(43,44);(1,2)(3,4)(5,6,9,8)(7,10,11,12)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,27,26,28)(29,31,34,30)(32,35,39,37)(33,38,41,36)(40,43,42,44);(5,7,9,11)(6,10,8,12)(13,16,18,24)(14,19,15,21)(17,26,25,23)(20,28,22,27)(29,32,34,39)(30,35,31,37)(33,42,41,40)(36,44,38,43);(5,8,9,6)(7,12,11,10)(13,17)(14,20)(15,22)(16,23)(18,25)(19,27)(21,28)(24,26)(29,33,34,41)(30,36,31,38)(32,40,39,42)(35,43,37,44)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=i,d^2=i,e^2=fgi,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, 3209756);