Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4452694 |
| automorphism | Aut(G) order 4194304 |
| 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 | |
1794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(8,12)(10,11)(16,23)(20,25)(21,27)(26,28)(29,30,33,36)(31,34,37,40)(32,35,39,42)(38,41,43,44);(2,3)(5,6,9,7)(8,10,12,11)(14,18)(19,24)(20,25)(26,28)(31,37)(34,40)(38,43)(41,44);(2,3)(5,7,9,6)(8,11,12,10)(13,14)(15,19)(16,20)(17,18)(21,26)(22,24)(23,25)(27,28)(29,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,44);(1,3,4,2)(5,8)(6,10)(7,11)(9,12)(13,15,17,22)(14,19,18,24)(16,21,23,27)(20,26,25,28)(29,31)(30,34)(32,38)(33,37)(35,41)(36,40)(39,43)(42,44);(13,16)(14,20)(15,21)(17,23)(18,25)(19,26)(22,27)(24,28)(29,32,33,39)(30,35,36,42)(31,38,37,43)(34,41,40,44)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=fi,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4363860);