Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1708 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)(6,7)(8,12)(14,18)(15,21)(20,25)(22,27)(29,30,34,37)(31,32,39,35)(33,41,43,36)(38,44,42,40);(6,7)(10,11)(15,21)(19,24)(22,27)(26,28)(29,31,34,39)(30,35,37,32)(33,42,43,38)(36,40,41,44);(5,6,9,7)(8,10,12,11)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,32,34,35)(30,31,37,39)(33,40,43,44)(36,38,41,42);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,31)(30,32)(33,38)(34,39)(35,37)(36,40)(41,44)(42,43);(5,8,9,12)(6,10,7,11)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33,34,43)(30,36,37,41)(31,38,39,42)(32,40,35,44)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=h,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4376519);