Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,9,11)(7,8,12,10)(14,18)(15,21)(20,25)(23,27)(29,30)(31,35)(32,34)(33,36)(37,39)(38,43)(40,42)(41,44);(5,6,9,11)(7,8,12,10)(16,24)(20,25)(23,27)(26,28)(30,32)(31,37)(36,40)(38,41);(5,7,9,12)(6,10,11,8)(13,14,17,18)(15,22,21,19)(16,20,24,25)(23,28,27,26)(29,31,34,37)(30,35,32,39)(33,41,42,38)(36,44,40,43);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,15,17,21)(14,19,18,22)(16,23,24,27)(20,26,25,28)(29,32,34,30)(31,35,37,39)(33,40,42,36)(38,43,41,44);(5,8)(6,7)(9,10)(11,12)(13,16)(14,20)(15,23)(17,24)(18,25)(19,26)(21,27)(22,28)(29,33,34,42)(30,36,32,40)(31,38,37,41)(35,43,39,44)Polycyclic presentation:
2^9|a^2=fh,b^2=fh,c^2=fg,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5124514);