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 | |
4636 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)(9,10,13,16)(11,17,19,14)(12,15,21,20)(18,23,24,22)(26,32)(27,35)(28,37)(34,36);(1,3,5,8)(2,6,7,4)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,26)(27,33)(28,38)(29,37)(30,34)(31,32)(35,40)(36,39);(1,3)(2,4)(5,8)(6,7)(25,27,31,35)(26,33,32,40)(28,30,37,39)(29,36,38,34);(1,4)(2,3)(5,6)(7,8)(9,10)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24)(25,28)(26,29)(27,30)(31,37)(32,38)(33,36)(34,40)(35,39);(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,29)(26,28)(27,36)(30,33)(31,38)(32,37)(34,35)(39,40);(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)(25,30)(26,34)(27,37)(28,35)(29,33)(31,39)(32,36)(38,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8378838);