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 | |
4327 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10,14,18)(11,17,13,15)(12,16,20,23)(19,24,21,22)(25,26)(27,29)(28,36)(30,32)(31,34)(33,39)(35,40)(37,38);(1,2,5,3)(4,6,8,7)(9,11,14,13)(10,15,18,17)(12,19,20,21)(16,22,23,24)(25,27,30,34)(26,31,32,29)(28,37,39,35)(33,38,36,40);(2,3)(6,7)(12,20)(16,23)(19,21)(22,24)(26,32)(27,34)(33,36)(35,37);(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,28,30,39)(26,33,32,36)(27,35,34,37)(29,38,31,40);(1,3,5,2)(4,7,8,6)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,24)(18,23)(25,26)(27,31)(28,33)(29,34)(30,32)(35,40)(36,39)(37,38);(1,4)(2,6)(3,7)(5,8)(9,13,14,11)(10,17,18,15)(12,21,20,19)(16,24,23,22)(25,29,30,31)(26,34,32,27)(28,38,39,40)(33,37,36,35)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8002080);