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 | |
4630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,15)(11,16,18,13)(12,20,21,17)(19,22,24,23)(25,26)(27,35)(28,39)(29,34)(30,32)(31,38)(33,37)(36,40);(1,3,5,8)(2,6,4,7)(9,11)(10,13)(12,19)(14,18)(15,16)(17,22)(20,23)(21,24)(25,27)(26,31)(28,36)(29,37)(30,38)(32,35)(33,40)(34,39);(10,15)(11,18)(12,21)(22,23)(26,32)(31,35)(33,39)(34,40);(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,22)(21,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,38)(36,40)(37,39);(1,2,5,4)(3,6,8,7)(9,12)(10,17)(11,19)(13,22)(14,21)(15,20)(16,23)(18,24)(25,28,30,37)(26,33,32,39)(27,36,38,29)(31,40,35,34);(1,4,5,2)(3,7,8,6)(9,13,14,16)(10,11,15,18)(12,22,21,23)(17,19,20,24)(25,29,30,36)(26,34,32,40)(27,37,38,28)(31,39,35,33)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,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, 7633797);