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 | |
4392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,19)(12,22)(13,18)(14,16)(15,21)(17,23)(20,24)(25,26)(27,34)(28,38)(29,35)(30,33)(31,32)(36,37)(39,40);(1,3,5,8)(2,6,4,7)(9,11,14,21)(10,15,16,19)(12,20,23,13)(17,24,22,18)(25,27)(26,29)(28,36)(30,37)(31,35)(32,34)(33,39)(38,40);(1,4,5,2)(3,7,8,6)(10,16)(15,19)(17,22)(18,24)(26,32)(27,35)(28,33)(37,40);(9,10)(11,15)(12,17)(13,18)(14,16)(19,21)(20,24)(22,23)(25,28,31,33)(26,30,32,38)(27,36,35,39)(29,37,34,40);(9,12)(10,17)(11,20)(13,21)(14,23)(15,24)(16,22)(18,19)(25,29)(26,27)(28,37)(30,36)(31,34)(32,35)(33,40)(38,39);(9,13,14,20)(10,18,16,24)(11,12,21,23)(15,17,19,22)(25,30)(26,33)(27,37)(28,32)(29,39)(31,38)(34,36)(35,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gh,d^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7853687);