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 | |
4448 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(11,17)(15,22)(18,20)(23,24)(25,26)(27,31)(28,32)(29,37)(30,34)(33,38)(35,36)(39,40);(1,3,5,8)(2,6,7,4)(9,10)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(25,27)(26,31)(28,34)(29,35)(30,32)(33,39)(36,37)(38,40);(1,3)(2,4)(5,8)(6,7)(10,14)(15,22)(16,19)(23,24)(25,26,30,34)(27,28,32,31)(29,37,38,33)(35,39,40,36);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,20)(13,17)(14,22)(16,24)(18,21)(19,23)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,39)(38,40);(9,12)(10,16)(11,18)(13,21)(14,19)(15,23)(17,20)(22,24)(25,28)(26,32)(27,34)(29,36)(30,31)(33,40)(35,37)(38,39);(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8832152);