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 | |
1710 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)(5,6,9,8)(7,10,11,12)(14,18)(15,21)(16,23)(26,28)(29,30)(31,37)(32,36)(33,34)(35,38)(39,43)(40,41)(42,44);(7,11)(10,12)(15,21)(19,24)(22,27)(26,28)(30,34)(31,38)(32,40)(42,43);(5,7,9,11)(6,10,8,12)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31)(30,35)(32,39)(33,38)(34,37)(36,42)(40,44)(41,43);(1,2)(3,4)(5,6,9,8)(7,10,11,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,30)(31,37)(32,41)(33,34)(35,38)(36,40)(39,42)(43,44);(5,8,9,6)(7,12,11,10)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,32)(30,36)(31,39)(33,40)(34,41)(35,42)(37,43)(38,44)Polycyclic presentation:
2^9|a^2=fh,c^2=gh,d^2=fh,e^2=fh,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, 2992002);