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 | |
4372 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)(9,10,14,12)(11,17,18,15)(13,16,22,20)(19,24,21,23);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,17)(13,21)(14,18)(16,24)(19,22)(20,23)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,36)(39,40);(1,3,5,4)(2,7,6,8)(10,12)(11,18)(16,20)(19,21)(26,31)(28,32)(34,37)(36,39);(9,12,14,10)(11,17,18,15)(13,20,22,16)(19,24,21,23)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,39,37,36);(1,3,5,4)(2,7,6,8)(9,11)(10,15)(12,17)(13,19)(14,18)(16,23)(20,24)(21,22)(25,28,30,32)(26,33,31,27)(29,36,38,39)(34,40,37,35);(1,4,5,3)(2,8,6,7)(9,13,14,22)(10,16,12,20)(11,19,18,21)(15,23,17,24)(25,29)(26,34)(27,35)(28,36)(30,38)(31,37)(32,39)(33,40)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gh,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7635441);