Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518920 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#459 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1783 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6,9,8)(7,10,12,11)(13,14)(15,23)(16,28)(17,19)(18,31)(20,24)(21,29)(22,25)(26,41)(27,34)(30,36)(32,39)(33,40)(35,42)(37,38)(43,44);(1,2,4,3)(5,7)(6,10)(8,11)(9,12)(14,19)(15,24)(16,29)(18,25)(20,34)(21,36)(23,32)(26,42)(27,39)(28,33)(30,40)(37,43);(5,8,9,6)(7,11,12,10)(13,15,25,39,17,27,18,24)(14,20,31,34,19,32,22,23)(16,26,40,44,30,42,29,38)(21,35,36,43,33,41,28,37);(7,12)(10,11)(13,16,18,29,17,30,25,40)(14,21,22,28,19,33,31,36)(15,26,24,38,27,42,39,44)(20,35,23,37,32,41,34,43);(1,3,4,2)(14,19)(16,30)(20,32)(22,31)(23,34)(26,42)(29,40)(38,44)Polycyclic presentation:
2^9|a^2=ghi,b^2=fi,c^2=hi,d^2=g,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7511440);