Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
1726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,8)(7,11,12,10)(13,14)(15,20)(16,21)(17,22)(18,19)(23,28)(24,25)(26,27)(30,34)(32,40)(35,37)(38,44);(2,4)(14,19)(17,20)(21,27)(25,28)(29,30,33,34)(31,37,39,35)(32,41,40,36)(38,42,44,43);(1,2)(3,4)(5,6,9,8)(7,10,12,11)(13,15,18,22)(14,20,19,17)(16,24,26,23)(21,25,27,28)(29,30)(31,37)(32,36)(33,34)(35,39)(38,43)(40,41)(42,44);(5,7,9,12)(6,10,8,11)(13,16,18,26)(14,21,19,27)(15,23,22,24)(17,25,20,28)(29,31,33,39)(30,35,34,37)(32,38,40,44)(36,42,41,43);(5,8,9,6)(7,11,12,10)(13,17,18,20)(14,22,19,15)(16,25,26,28)(21,24,27,23)(29,32,33,40)(30,36,34,41)(31,38,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=fgi,d^2=fi,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1688210);