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 | |
4525 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,15)(12,18)(13,21)(14,16)(17,20)(19,24)(22,23)(26,31)(27,34)(29,37)(36,39);(2,3)(6,7)(9,11,14,20)(10,15,16,17)(12,21,23,19)(13,22,24,18)(25,26,30,31)(27,28,34,32)(29,33,37,38)(35,36,40,39);(1,3,5,2)(4,7,8,6)(10,16)(11,20)(18,22)(19,21)(25,27,30,34)(26,32,31,28)(29,35,37,40)(33,39,38,36);(1,4)(2,6)(3,7)(5,8)(9,11)(10,17)(12,21)(13,22)(14,20)(15,16)(18,24)(19,23)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,40)(38,39);(9,12)(10,18)(11,21)(13,15)(14,23)(16,22)(17,24)(19,20)(25,28)(26,27)(29,36)(30,32)(31,34)(33,35)(37,39)(38,40);(9,13,14,24)(10,19,16,21)(11,22,20,18)(12,17,23,15)(25,29,30,37)(26,33,31,38)(27,35,34,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9844951);