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 | |
1768 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)(14,18)(15,21)(16,23)(26,28)(29,30)(31,34)(32,37)(33,36)(35,39)(38,41)(40,44)(42,43);(5,7,10,9)(6,11,8,12)(15,21)(19,24)(22,27)(26,28)(30,31)(32,39)(33,41)(43,44);(5,8,10,6)(7,11,9,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31,34,30)(32,35,39,37)(33,38,41,36)(40,43,42,44);(1,2)(3,4)(5,9,10,7)(6,11,8,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,32,34,39)(30,35,31,37)(33,42,41,40)(36,44,38,43);(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33,34,41)(30,36,31,38)(32,40,39,42)(35,43,37,44)Polycyclic presentation:
2^9|a^2=fh,b^2=fh,c^2=h,d^2=h,e^2=g,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, 2984396);