Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | C22×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1747 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)(15,21)(16,30)(17,36)(18,24)(19,25)(20,26)(22,34)(23,29)(27,39)(28,38)(31,37)(32,42)(33,41)(35,40)(43,44);(1,2,4,3)(5,7)(6,10)(8,11)(9,12)(13,15)(14,21)(16,31)(17,18)(19,28)(20,29)(22,39)(23,26)(24,36)(25,38)(27,34)(30,37)(32,35)(33,43)(40,42)(41,44);(5,8,9,6)(7,11,12,10)(13,16,19,34,20,35,18,33)(14,22,24,30,26,41,25,40)(15,27,17,31,29,43,28,32)(21,37,38,39,23,42,36,44);(7,12)(10,11)(13,17,19,29,20,28,18,15)(14,23,24,36,26,21,25,38)(16,32,34,27,35,31,33,43)(22,37,30,39,41,42,40,44);(1,3,4,2)(13,18,20,19)(14,24,26,25)(15,17,29,28)(16,33,35,34)(21,36,23,38)(22,30,41,40)(27,31,43,32)(37,44,42,39)Polycyclic presentation:
2^9|a^2=gh,b^2=f,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,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7511465);