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 | |
4614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,13)(12,20)(14,15)(16,18)(17,22)(19,21)(23,24)(25,26)(27,29)(28,33)(30,35)(31,36)(32,34)(37,39)(38,40);(1,2,5,6)(3,7,4,8)(9,10,14,15)(11,16,18,13)(12,17,22,20)(19,21,24,23)(25,27,31,34)(26,32,36,29)(28,38,39,35)(30,37,40,33);(10,15)(11,18)(17,20)(19,24)(25,28,31,39)(26,33,36,37)(27,35,34,38)(29,30,32,40);(1,3,5,4)(2,7,6,8)(9,11)(10,16)(12,19)(13,15)(14,18)(17,21)(20,23)(22,24)(25,29,31,32)(26,34,36,27)(28,30,39,40)(33,38,37,35);(9,12)(10,17)(11,19)(13,23)(14,22)(15,20)(16,21)(18,24)(25,30)(26,35)(27,37)(28,32)(29,39)(31,40)(33,34)(36,38);(1,4,5,3)(2,8,6,7)(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,24,20,19)(25,26)(27,32)(28,33)(29,34)(30,38)(31,36)(35,40)(37,39)Polycyclic presentation:
2^9|b^2=g,c^2=hi,d^2=gh,f^2=ghi,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, 7799221);