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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4608 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)(3,4)(5,6)(7,8)(11,17)(12,20)(15,18)(16,21)(25,26)(27,29)(28,36)(30,32)(31,39)(33,34)(35,40)(37,38);(2,6)(4,7)(9,10)(11,18)(12,21)(13,14)(15,17)(16,20)(19,23)(22,24)(25,26,30,32)(27,33,34,29)(28,36,39,31)(35,38,37,40);(1,2,5,6)(3,7,8,4)(10,14)(11,17)(16,21)(19,22)(27,34)(29,33)(35,37)(38,40);(1,3)(2,7)(4,6)(5,8)(9,11)(10,15)(12,22)(13,17)(14,18)(16,24)(19,20)(21,23)(25,27)(26,29)(28,37)(30,34)(31,40)(32,33)(35,39)(36,38);(1,4,5,7)(2,3,6,8)(9,12)(10,16)(11,19)(13,20)(14,21)(15,23)(17,22)(18,24)(25,28,30,39)(26,31,32,36)(27,35,34,37)(29,38,33,40);(1,3)(2,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,21)(22,24)(25,29)(26,27)(28,38)(30,33)(31,35)(32,34)(36,37)(39,40)Polycyclic presentation:
2^9|b^2=gh,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8840301);