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 | |
4531 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)(11,17)(15,18)(19,21)(23,24)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,39)(37,38);(2,3)(6,7)(9,10)(11,18)(12,20)(13,14)(15,17)(16,22)(19,23)(21,24)(26,29)(32,34)(33,37)(39,40);(1,3,5,2)(4,7,8,6)(10,14)(15,18)(16,20)(23,24)(25,27,31,28)(26,32,29,34)(30,36,38,35)(33,40,37,39);(1,4)(2,6)(3,7)(5,8)(9,11,13,17)(10,15,14,18)(12,21,22,19)(16,24,20,23)(25,28,31,27)(26,32,29,34)(30,36,38,35)(33,39,37,40);(9,12,13,22)(10,16,14,20)(11,19,17,21)(15,23,18,24)(25,29,31,26)(27,34,28,32)(30,37,38,33)(35,40,36,39);(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,30,31,38)(26,33,29,37)(27,35,28,36)(32,39,34,40)Polycyclic presentation:
2^9|a^2=gh,c^2=g,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8323983);