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 | |
1778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(8,12)(10,11)(14,18)(15,21)(20,25)(22,27)(30,34)(31,37)(32,39)(42,44);(2,3)(5,6,9,7)(8,10,12,11)(15,21)(19,24)(22,27)(26,28)(30,34)(31,37)(36,41)(38,43);(2,3)(5,7,9,6)(8,11,12,10)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,30)(31,35)(32,36)(33,34)(37,40)(38,42)(39,41)(43,44);(1,3,4,2)(5,8)(6,10)(7,11)(9,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,31,33,37)(30,35,34,40)(32,38,39,43)(36,42,41,44);(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,32,33,39)(30,36,34,41)(31,38,37,43)(35,42,40,44)Polycyclic presentation:
2^9|a^2=f,b^2=fgi,c^2=fgi,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, 4363876);