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 | |
1789 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(7,11)(9,12)(14,18)(16,23)(19,24)(22,27)(29,30,34,35)(31,36,37,32)(33,41,42,38)(39,40,44,43);(2,4)(5,6,8,10)(7,9,11,12)(14,18)(15,21)(20,25)(22,27)(30,35)(31,37)(33,42)(40,43);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31)(30,36)(32,35)(33,39)(34,37)(38,43)(40,41)(42,44);(5,7,8,11)(6,9,10,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,32,34,36)(30,37,35,31)(33,43,42,40)(38,39,41,44);(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,33,34,42)(30,38,35,41)(31,39,37,44)(32,40,36,43)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=g,d^2=i,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, 3059012);