Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#546 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,17)(12,18)(13,19)(14,16)(15,21)(20,23)(22,24)(25,26)(27,32)(28,30)(29,36)(31,34)(33,39)(35,37)(38,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,20)(13,18)(14,21)(16,17)(19,22)(23,24)(25,27)(26,32)(28,36)(29,30)(31,35)(33,38)(34,37)(39,40);(1,3,5,4)(2,6,8,7)(10,16)(13,23)(15,17)(18,24)(27,35)(29,39)(32,37)(33,36);(1,3,5,4)(2,6,8,7)(9,11)(10,17)(12,19)(13,18)(14,21)(15,16)(20,22)(23,24)(25,28,31,38)(26,30,34,40)(27,33,35,36)(29,32,39,37);(9,12)(10,18)(11,20)(13,15)(14,22)(16,24)(17,23)(19,21)(25,29)(26,33)(27,30)(28,37)(31,39)(32,38)(34,36)(35,40);(1,4,5,3)(2,7,8,6)(9,13,14,23)(10,19,16,20)(11,18,21,24)(12,15,22,17)(25,30)(26,28)(27,29)(31,40)(32,36)(33,37)(34,38)(35,39)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=ghi,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7995530);