Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| 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×#3321 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,18,23)(15,20,27,37)(16,29,31,21)(17,32,34,22)(19,24,35,41)(25,43,40,33)(26,44,39,30)(28,38,36,42);(5,7,9,8)(6,10,12,11)(13,15,18,27)(14,20,23,37)(16,30,31,44)(17,33,34,43)(19,36,35,28)(21,39,29,26)(22,40,32,25)(24,42,41,38);(2,4)(5,7,9,8)(6,11,12,10)(13,16,18,31)(14,21,23,29)(15,25,27,40)(17,24,34,41)(19,22,35,32)(20,33,37,43)(26,42,39,38)(28,44,36,30);(5,8,9,7)(6,10,12,11)(16,22)(17,29)(19,35)(21,34)(24,41)(25,44)(26,33)(28,36)(30,40)(31,32)(38,42)(39,43);(13,17,18,34)(14,22,23,32)(15,26,27,39)(16,24,31,41)(19,21,35,29)(20,30,37,44)(25,42,40,38)(28,43,36,33)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=fi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6484355);