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 | |
4589 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,8,7)(9,10)(11,15)(12,13)(14,17)(16,20)(18,23)(19,21)(22,24)(28,36)(32,37)(35,40)(38,39);(2,6)(4,7)(9,11,14,20)(10,15,17,16)(12,21,22,18)(13,19,24,23)(25,26)(27,31)(28,37)(29,33)(30,34)(32,36)(35,38)(39,40);(9,11)(10,16)(12,18)(13,19)(14,20)(15,17)(21,22)(23,24)(25,27)(26,29)(28,35)(30,33)(31,34)(32,38)(36,40)(37,39);(1,3,5,8)(2,7,6,4)(9,11)(10,15)(12,18)(13,23)(14,20)(16,17)(19,24)(21,22)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,40)(37,38);(1,4)(2,3)(5,7)(6,8)(9,12)(10,13)(11,18)(14,22)(15,23)(16,19)(17,24)(20,21)(25,28,30,36)(26,32,34,37)(27,35,33,40)(29,38,31,39);(1,3)(2,4)(5,8)(6,7)(9,13)(10,12)(11,19)(14,24)(15,21)(16,18)(17,22)(20,23)(25,29,30,31)(26,33,34,27)(28,38,36,39)(32,40,37,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8413504);