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 | |
4794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,14,15)(11,12,16,19)(13,17,21,22)(18,20,23,24)(25,26,30,32)(27,34,37,31)(28,33,38,40)(29,36,39,35);(2,6)(3,7)(10,15)(12,19)(17,22)(20,24)(25,27)(26,31)(28,36)(29,33)(30,37)(32,34)(35,38)(39,40);(1,3)(2,4)(5,7)(6,8)(13,21)(17,22)(18,23)(20,24)(26,32)(29,39)(31,34)(33,40);(1,4)(2,3)(5,8)(6,7)(9,11)(10,12)(13,18)(14,16)(15,19)(17,20)(21,23)(22,24)(25,28,30,38)(26,33,32,40)(27,36,37,35)(29,34,39,31);(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,22)(21,24)(25,27)(26,34)(28,35)(29,33)(30,37)(31,32)(36,38)(39,40);(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,29)(26,35)(27,33)(28,34)(30,39)(31,38)(32,36)(37,40)Polycyclic presentation:
2^9|a^2=g,d^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8832349);