Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1770 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)(6,9)(7,11)(13,14,17,20)(15,21,22,18)(16,24,25,19)(23,26,28,27)(29,30,33,36)(31,37,39,34)(32,40,41,35)(38,42,44,43);(15,22)(18,21)(23,28)(26,27)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(5,6,8,9)(7,10,11,12)(13,15,17,22)(14,18,20,21)(16,23,25,28)(19,26,24,27)(29,31)(30,34)(32,38)(33,39)(35,42)(36,37)(40,43)(41,44);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,14)(15,18)(16,19)(17,20)(21,22)(23,26)(24,25)(27,28)(29,30)(31,34)(32,40)(33,36)(35,41)(37,39)(38,43)(42,44);(13,16,17,25)(14,19,20,24)(15,23,22,28)(18,26,21,27)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fg,d^2=gi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4342363);