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 | |
4394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,15)(11,16,18,13)(12,20,22,17)(19,21,24,23)(25,26,31,30)(27,35,36,32)(28,33,38,39)(29,34,37,40);(1,2,4,5)(3,6,7,8)(25,27)(26,32)(28,37)(29,38)(30,35)(31,36)(33,34)(39,40);(10,15)(11,18)(17,20)(19,24)(27,36)(29,37)(32,35)(34,40);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,28,31,38)(26,33,30,39)(27,37,36,29)(32,34,35,40);(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,29,31,37)(26,34,30,40)(27,38,36,28)(32,33,35,39);(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,19,20,24)(25,30,31,26)(27,32,36,35)(28,39,38,33)(29,40,37,34)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,d^2=g,e^2=hi,f^2=hi,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, 7640016);