Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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 | |
1754 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)(5,6,9,12)(7,10,8,11)(13,14)(15,21)(16,18)(17,24)(19,22)(20,26)(23,27)(25,28)(29,30,33,36)(31,37,38,34)(32,40,41,35)(39,42,44,43);(5,7,9,8)(6,10,12,11)(14,16)(15,22)(20,24)(23,25)(31,38)(34,37)(39,44)(42,43);(5,6)(7,10)(8,11)(9,12)(13,15)(14,19)(16,21)(17,25)(18,22)(20,28)(23,26)(24,27)(29,31,33,38)(30,34,36,37)(32,39,41,44)(35,42,40,43);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,16,18,14)(15,21,22,19)(17,24,26,20)(23,28,25,27)(29,30)(31,34)(32,35)(33,36)(37,38)(39,42)(40,41)(43,44);(5,8,9,7)(6,11,12,10)(13,17)(14,20)(15,23)(16,24)(18,26)(19,27)(21,28)(22,25)(29,32,33,41)(30,35,36,40)(31,39,38,44)(34,42,37,43)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=fh,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4287899);