Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7780 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)(6,10)(7,11)(14,19)(17,30)(20,24)(21,27)(26,36)(28,41)(32,37)(39,43);(5,6,9,10)(7,8,11,12)(14,19)(15,23)(21,27)(22,31)(25,35)(26,36)(34,40)(39,43);(2,3)(5,7)(6,8)(9,11)(10,12)(13,14)(15,24)(16,27)(17,22)(18,19)(20,23)(21,29)(25,32)(26,38)(28,40)(30,31)(33,36)(34,41)(35,37)(39,42)(43,44);(1,3,4,2)(5,8)(6,11)(7,10)(9,12)(13,15,18,23)(14,20,19,24)(16,25,29,35)(17,26,30,36)(21,32,27,37)(22,33,31,38)(28,39,41,43)(34,42,40,44);(13,16,18,29)(14,21,19,27)(15,25,23,35)(17,28,30,41)(20,32,24,37)(22,34,31,40)(26,39,36,43)(33,42,38,44);(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=ghi,b^2=g,d^2=gh,e^2=i,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7581838);