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 | |
4465 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)(3,4)(5,6)(7,8)(9,10,14,18)(11,16,12,15)(13,17,22,21)(19,24,20,23)(25,26,30,32)(27,33,35,29)(28,31,37,38)(34,39,40,36);(2,6)(3,7)(9,11,14,12)(10,15,18,16)(13,19,22,20)(17,23,21,24)(25,27,30,35)(26,29,32,33)(28,34,37,40)(31,36,38,39);(1,3,5,7)(2,4,6,8)(25,26)(27,29)(28,31)(30,32)(33,35)(34,36)(37,38)(39,40);(1,4)(2,7)(3,6)(5,8)(9,10)(11,15)(12,16)(13,21)(14,18)(17,22)(19,24)(20,23)(25,26)(27,33)(28,31)(29,35)(30,32)(34,39)(36,40)(37,38);(9,12,14,11)(10,16,18,15)(13,20,22,19)(17,24,21,23)(25,28,30,37)(26,31,32,38)(27,34,35,40)(29,36,33,39);(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,24)(18,21)(25,29,30,33)(26,27,32,35)(28,36,37,39)(31,34,38,40)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=i,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8364248);