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 | |
4495 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,18)(17,23)(19,20)(26,31)(27,33)(29,37)(36,40);(1,2)(3,8)(4,6)(5,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,24)(20,21)(22,23)(25,26,30,31)(27,32,33,28)(29,34,37,39)(35,40,38,36);(1,3)(2,4)(5,6)(7,8)(11,18)(13,16)(19,20)(21,24)(25,27,30,33)(26,32,31,28)(29,38,37,35)(34,36,39,40);(3,6)(4,8)(9,11,14,18)(10,16,15,13)(12,20,22,19)(17,21,23,24)(25,27)(26,32)(28,31)(29,35)(30,33)(34,40)(36,39)(37,38);(1,4,5,8)(2,6,7,3)(9,12,14,22)(10,17,15,23)(11,19,18,20)(13,21,16,24)(25,28,30,32)(26,33,31,27)(29,36,37,40)(34,38,39,35);(1,2)(3,8)(4,6)(5,7)(9,13,14,16)(10,18,15,11)(12,21,22,24)(17,20,23,19)(25,29)(26,34)(27,35)(28,36)(30,37)(31,39)(32,40)(33,38)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=gh,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8821387);