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 | |
4461 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,20)(13,17)(14,15)(16,21)(19,23)(22,24)(25,26,31,29)(27,28,35,32)(30,33,38,37)(34,36,40,39);(1,2,5,4)(3,7,8,6)(10,15)(11,17)(12,21)(23,24)(25,27,31,35)(26,32,29,28)(30,34,38,40)(33,39,37,36);(2,4)(6,7)(10,15)(13,18)(16,20)(23,24)(30,38)(33,37)(34,40)(36,39);(3,8)(6,7)(9,11,14,17)(10,13,15,18)(12,22,21,19)(16,24,20,23)(25,28)(26,27)(29,35)(30,36)(31,32)(33,34)(37,40)(38,39);(1,3)(2,6)(4,7)(5,8)(9,12)(10,16)(11,19)(13,23)(14,21)(15,20)(17,22)(18,24)(25,29,31,26)(27,32,35,28)(30,37,38,33)(34,39,40,36);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,15)(12,23)(14,18)(16,22)(19,20)(21,24)(25,30,31,38)(26,33,29,37)(27,34,35,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,d^2=h,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8282714);