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 | C25 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 44 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(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);(5,6,8,10)(7,9,12,11)(14,19)(16,27)(20,23)(22,29)(24,37)(28,40)(33,36)(38,44);(5,6)(7,9)(8,10)(11,12)(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);(5,7,8,12)(6,9,10,11)(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);(13,16,18,27)(14,21,19,31)(15,24,26,37)(17,28,30,40)(20,32,23,35)(22,34,29,41)(25,38,39,44)(33,42,36,43);(13,17)(14,22)(15,25)(16,28)(18,30)(19,29)(20,33)(21,34)(23,36)(24,38)(26,39)(27,40)(31,41)(32,42)(35,43)(37,44)Polycyclic presentation:
2^9|a^2=i,b^2=gh,d^2=g,e^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, 7533571);