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 | |
1722 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)(7,10,12,11)(14,18)(15,21)(16,23)(26,28)(29,30)(31,36)(32,40)(33,38)(34,35)(37,43)(39,42)(41,44);(5,7,9,12)(6,10,8,11)(14,18)(15,21)(20,25)(22,27)(30,35)(33,36)(37,40)(42,44);(5,7)(6,10)(8,11)(9,12)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31,34,38)(30,36,35,33)(32,41,43,39)(37,42,40,44);(1,2,3,4)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,32,34,43)(30,37,35,40)(31,39,38,41)(33,42,36,44);(5,8,9,6)(7,11,12,10)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33,34,36)(30,38,35,31)(32,42,43,44)(37,41,40,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=fg,d^2=gi,e^2=fh,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, 1620947);