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 | |
4622 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)(3,6)(4,7)(5,8)(12,20)(15,22)(18,24)(21,23)(25,26)(27,29)(28,36)(30,33)(31,35)(32,37)(34,39)(38,40);(1,3)(2,4)(5,7)(6,8)(9,10)(11,16)(12,15)(13,17)(14,19)(18,23)(20,22)(21,24)(28,37)(32,36)(34,40)(38,39);(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27)(26,31)(28,34)(29,33)(30,35)(32,39)(36,38)(37,40);(9,12)(10,15)(11,18)(13,20)(14,21)(16,23)(17,22)(19,24)(25,28)(26,32)(27,34)(29,38)(30,37)(31,39)(33,36)(35,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,24)(21,22)(25,29,30,31)(26,27,33,35)(28,38,37,39)(32,34,36,40);(1,4,5,6)(2,7,8,3)(9,10)(11,14)(12,15)(13,17)(16,19)(18,21)(20,22)(23,24)(25,27)(26,29)(28,34)(30,35)(31,33)(32,38)(36,39)(37,40)Polycyclic presentation:
2^9|e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dhi,c^b=ci,d^b=dh,f^e=fghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8004947);