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 | |
4427 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,13)(12,16)(14,15)(17,21)(18,20)(19,22)(23,24)(25,26)(27,32)(28,31)(29,33)(30,38)(34,40)(35,36)(37,39);(2,3)(6,7)(12,19)(16,22)(18,23)(20,24)(25,27)(26,32)(28,36)(29,34)(30,37)(31,35)(33,40)(38,39);(1,3,5,2)(4,7,8,6)(10,15)(11,17)(16,22)(18,23)(26,28)(27,35)(33,37)(34,38);(1,4)(2,6)(3,7)(5,8)(9,11,14,17)(10,13,15,21)(12,18,19,23)(16,20,22,24)(25,28,31,26)(27,36,35,32)(29,37,39,33)(30,38,40,34);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29)(26,33)(27,34)(28,37)(30,36)(31,39)(32,40)(35,38);(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,30)(26,34)(27,37)(28,38)(29,32)(31,40)(33,35)(36,39)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9817496);