Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62862 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 44 points:
(1,2,4,3)(5,6,9,11)(7,8,12,10)(13,14,18,19)(15,23,26,20)(16,21,27,31)(17,29,30,22)(24,35,37,32)(25,33,39,36)(28,41,40,34)(38,42,44,43);(5,7,9,12)(6,10,11,8)(14,19)(16,27)(20,23)(22,29)(24,37)(28,40)(33,36)(38,44);(5,7)(6,8)(9,12)(10,11)(13,14)(15,23)(16,21)(17,22)(18,19)(20,26)(24,35)(25,36)(27,31)(28,34)(29,30)(32,37)(33,39)(38,43)(40,41)(42,44);(5,8)(6,7)(9,10)(11,12)(13,15,18,26)(14,20,19,23)(16,24,27,37)(17,25,30,39)(21,32,31,35)(22,33,29,36)(28,38,40,44)(34,42,41,43);(1,3,4,2)(13,16)(14,21)(15,24)(17,28)(18,27)(19,31)(20,32)(22,34)(23,35)(25,38)(26,37)(29,41)(30,40)(33,42)(36,43)(39,44);(13,17,18,30)(14,22,19,29)(15,25,26,39)(16,28,27,40)(20,33,23,36)(21,34,31,41)(24,38,37,44)(32,42,35,43)Polycyclic presentation:
2^9|a^2=i,b^2=gh,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh,d^c=dh140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533757);