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 | |
4674 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)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,26)(27,33)(28,32)(29,34)(30,31)(35,40)(36,38)(37,39);(1,3,5,8)(2,6,4,7)(12,19)(16,22)(18,23)(20,24)(25,27,30,34)(26,29,31,33)(28,36,37,35)(32,40,39,38);(10,15)(11,17)(16,22)(18,23)(26,31)(27,34)(28,37)(38,40);(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,27)(26,29)(28,35)(30,34)(31,33)(32,38)(36,37)(39,40);(1,4,5,2)(3,7,8,6)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,28,30,37)(26,32,31,39)(27,35,34,36)(29,38,33,40);(9,13)(10,11)(12,20)(14,21)(15,17)(16,18)(19,24)(22,23)(25,29)(26,27)(28,38)(30,33)(31,34)(32,35)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=di,e^b=eh,f^b=fi,d^c=dh,e^c=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7896975);