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 | |
1841 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,8,10)(7,9,11,12)(13,14,17,20)(15,21,22,18)(16,24,25,19)(23,26,28,27)(29,30,34,35)(31,32,39,36)(33,41,42,37)(38,43,44,40);(2,4)(5,7,8,11)(6,9,10,12)(15,22)(18,21)(23,28)(26,27)(30,35)(32,36)(33,42)(38,44);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,14)(15,18)(16,19)(17,20)(21,22)(23,26)(24,25)(27,28)(29,31,34,39)(30,36,35,32)(33,38,42,44)(37,43,41,40);(5,6)(7,9)(8,10)(11,12)(13,15,17,22)(14,18,20,21)(16,23,25,28)(19,26,24,27)(29,32,34,36)(30,31,35,39)(33,40,42,43)(37,38,41,44);(13,16,17,25)(14,19,20,24)(15,23,22,28)(18,26,21,27)(29,33)(30,37)(31,38)(32,40)(34,42)(35,41)(36,43)(39,44)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=f,d^2=g,e^2=fg,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, 1718300);