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 | |
4855 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,17)(12,20)(13,18)(14,15)(16,21)(19,23)(22,24)(25,26,30,31)(27,33,34,32)(28,37,39,29)(35,40,38,36);(2,6)(3,8)(10,15)(12,21)(13,17)(19,22)(26,31)(29,37)(32,33)(36,40);(1,3)(2,4)(5,8)(6,7)(10,15)(11,18)(16,20)(19,22)(27,34)(32,33)(35,38)(36,40);(1,4)(2,3)(5,7)(6,8)(9,11)(10,13)(12,22)(14,18)(15,17)(16,24)(19,21)(20,23)(25,27,30,34)(26,32,31,33)(28,38,39,35)(29,36,37,40);(1,2)(3,4)(5,6)(7,8)(9,12)(10,16)(11,19)(13,23)(14,21)(15,20)(17,24)(18,22)(25,28)(26,29)(27,35)(30,39)(31,37)(32,40)(33,36)(34,38);(1,4)(2,3)(5,7)(6,8)(9,13)(10,11)(12,23)(14,17)(15,18)(16,19)(20,22)(21,24)(25,29)(26,28)(27,36)(30,37)(31,39)(32,38)(33,35)(34,40)Polycyclic presentation:
2^9|a^2=gh,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8731934);