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 | |
4549 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)(5,8)(6,7)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,26,30,31)(27,32,34,29)(28,33,36,39)(35,37,40,38);(1,3)(2,4)(5,6)(7,8)(12,19)(15,17)(20,24)(22,23)(26,31)(27,34)(28,36)(37,38);(1,4)(2,6)(3,8)(5,7)(10,11)(15,17)(16,18)(22,23);(1,2,5,8)(3,7,6,4)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,27)(26,32)(28,35)(29,31)(30,34)(33,37)(36,40)(38,39);(1,4)(2,3)(5,7)(6,8)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,28)(26,33)(27,35)(29,38)(30,36)(31,39)(32,37)(34,40);(1,3)(2,7)(4,8)(5,6)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,32)(26,27,31,34)(28,37,36,38)(33,40,39,35)Polycyclic presentation:
2^9|a^2=ghi,d^2=h,f^2=g,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, 9843625);