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 | |
4471 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)(3,7)(4,8)(5,6)(9,10,14,12)(11,15,17,19)(13,16,21,20)(18,22,23,24)(25,26)(27,34)(28,31)(29,37)(30,32)(33,38)(35,39)(36,40);(1,2,5,6)(3,7,8,4)(10,12)(15,19)(16,20)(22,24)(25,27,30,28)(26,31,32,34)(29,35,38,36)(33,39,37,40);(1,3)(2,4)(5,8)(6,7)(11,17)(13,21)(15,19)(16,20)(26,32)(29,38)(31,34)(35,36);(1,4)(2,3)(5,7)(6,8)(9,11,14,17)(10,15,12,19)(13,18,21,23)(16,22,20,24)(25,26)(27,31)(28,34)(29,33)(30,32)(35,39)(36,40)(37,38);(1,2)(3,4)(5,6)(7,8)(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24)(25,28,30,27)(26,31,32,34)(29,36,38,35)(33,39,37,40);(1,4)(2,3)(5,7)(6,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,37)(34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=gh,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8838292);