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 | |
1741 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)(5,6,10,8)(7,11,9,12)(13,14)(15,18)(16,21)(17,20)(19,23)(22,25)(24,28)(26,27)(29,30,34,37)(31,38,35,32)(33,41,42,36)(39,40,43,44);(5,7,10,9)(6,11,8,12)(14,15)(16,23)(17,25)(27,28)(31,35)(32,38)(39,43)(40,44);(5,8,10,6)(7,11,9,12)(13,15,18,14)(16,19,23,21)(17,22,25,20)(24,27,26,28)(29,31,34,35)(30,32,37,38)(33,39,42,43)(36,40,41,44);(1,2)(3,4)(5,9,10,7)(6,11,8,12)(13,16,18,23)(14,19,15,21)(17,26,25,24)(20,28,22,27)(29,32)(30,35)(31,37)(33,40)(34,38)(36,43)(39,41)(42,44);(13,17,18,25)(14,20,15,22)(16,24,23,26)(19,27,21,28)(29,33)(30,36)(31,39)(32,40)(34,42)(35,43)(37,41)(38,44)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=h,d^2=fgh,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, 2984322);