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×#525 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6680 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 44 points:
(1,2,4,3)(5,6,9,10)(7,8,11,12)(14,19)(15,23)(17,30)(21,27)(25,35)(28,41)(33,38)(42,44);(6,10)(7,11)(13,14,18,19)(15,20,23,24)(16,27,29,21)(17,22,30,31)(25,37,35,32)(26,33,36,38)(28,40,41,34)(39,44,43,42);(5,7)(6,8)(9,11)(10,12)(13,14)(15,24)(16,21)(17,22)(18,19)(20,23)(25,37)(26,38)(27,29)(28,34)(30,31)(32,35)(33,36)(39,44)(40,41)(42,43);(5,8)(6,7)(9,12)(10,11)(13,15)(14,20)(16,25)(17,26)(18,23)(19,24)(21,32)(22,33)(27,37)(28,39)(29,35)(30,36)(31,38)(34,42)(40,44)(41,43);(1,3,4,2)(13,16)(14,21)(15,25)(17,28)(18,29)(19,27)(20,32)(22,34)(23,35)(24,37)(26,39)(30,41)(31,40)(33,42)(36,43)(38,44);(13,17,18,30)(14,22,19,31)(15,26,23,36)(16,28,29,41)(20,33,24,38)(21,34,27,40)(25,39,35,43)(32,42,37,44)Polycyclic presentation:
2^9|a^2=i,b^2=h,e^2=ghi,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, 7533763);