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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(7,11)(9,12)(13,14,18,15)(16,19,24,21)(17,20,25,22)(23,26,28,27)(29,30,35,38)(31,33,32,36)(34,42,43,37)(39,44,40,41);(2,4)(14,15)(17,25)(19,21)(23,28)(29,31,35,32)(30,36,38,33)(34,40,43,39)(37,41,42,44);(1,2)(3,4)(5,6,8,10)(7,9,11,12)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,26,28,27)(29,32,35,31)(30,33,38,36)(34,40,43,39)(37,41,42,44);(5,7,8,11)(6,9,10,12)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27)(29,33)(30,32)(31,38)(34,41)(35,36)(37,40)(39,42)(43,44);(13,17)(14,20)(15,22)(16,23)(18,25)(19,26)(21,27)(24,28)(29,34,35,43)(30,37,38,42)(31,39,32,40)(33,41,36,44)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fi,d^2=fgi,e^2=g,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, 1459407);