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 | |
4413 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,8)(4,7)(5,6)(9,10)(11,18)(12,16)(13,17)(14,15)(19,24)(20,23)(21,22)(25,26,31,29)(27,32,35,37)(28,33,38,39)(30,36,40,34);(1,3)(2,4)(5,8)(6,7)(10,15)(11,17)(12,20)(22,24)(25,27)(26,32)(28,34)(29,37)(30,33)(31,35)(36,38)(39,40);(2,5)(4,8)(11,17)(12,20)(13,18)(16,23)(26,29)(27,35)(28,38)(30,40);(1,2,6,5)(3,8,7,4)(9,11,14,17)(10,13,15,18)(12,21,20,19)(16,24,23,22)(25,28)(26,33)(27,36)(29,39)(30,37)(31,38)(32,40)(34,35);(1,4,6,8)(2,7,5,3)(9,12,14,20)(10,16,15,23)(11,19,17,21)(13,22,18,24)(25,29,31,26)(27,37,35,32)(28,33,38,39)(30,36,40,34);(1,5,6,2)(3,8,7,4)(9,13)(10,17)(11,15)(12,22)(14,18)(16,21)(19,23)(20,24)(25,30,31,40)(26,34,29,36)(27,33,35,39)(28,37,38,32)Polycyclic presentation:
2^9|a^2=i,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8970476);