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 | |
1750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,11)(10,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,30,34,38)(31,36,32,35)(33,41,42,37)(39,43,40,44);(6,9)(10,12)(14,18)(16,24)(19,21)(22,28)(29,31,34,32)(30,35,38,36)(33,40,42,39)(37,44,41,43);(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,32,34,31)(30,36,38,35)(33,40,42,39)(37,44,41,43);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27)(29,30)(31,36)(32,35)(33,37)(34,38)(39,44)(40,43)(41,42);(13,16,17,24)(14,20,18,25)(15,22,23,28)(19,26,21,27)(29,33,34,42)(30,37,38,41)(31,39,32,40)(35,43,36,44)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,d^2=h,e^2=f,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, 1480257);