Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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, prime radical, cosocle | C24 | C25 |
| 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) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,9,12)(7,10,8,11)(13,14)(15,19)(16,23)(17,18)(20,25)(21,26)(22,28)(24,27)(29,30,34,32)(31,37,38,35)(33,40,41,36)(39,42,44,43);(5,7,9,8)(6,10,12,11)(14,18)(15,21)(20,23)(22,24)(31,38)(35,37)(39,44)(42,43);(5,6)(7,10)(8,11)(9,12)(13,15,17,21)(14,19,18,26)(16,24,25,22)(20,28,23,27)(29,31)(30,35)(32,37)(33,39)(34,38)(36,42)(40,43)(41,44);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14)(15,19)(16,20)(17,18)(21,26)(22,27)(23,25)(24,28)(29,32,34,30)(31,37,38,35)(33,40,41,36)(39,43,44,42);(5,8,9,7)(6,11,12,10)(13,16)(14,20)(15,22)(17,25)(18,23)(19,27)(21,24)(26,28)(29,33,34,41)(30,36,32,40)(31,39,38,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=f,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4288444);