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 | |
4593 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)(3,7)(4,8)(5,6)(9,10)(11,17)(12,21)(13,23)(14,16)(15,20)(18,19)(22,24)(25,26,32,30)(27,31,35,33)(28,34,29,39)(36,38,37,40);(2,6)(3,8)(9,11)(10,15)(12,22)(13,24)(14,20)(16,17)(18,23)(19,21)(25,27,32,35)(26,33,30,31)(28,36,29,37)(34,40,39,38);(1,3)(2,4)(5,8)(6,7)(10,16)(13,21)(15,17)(19,24)(25,28,32,29)(26,34,30,39)(27,36,35,37)(31,38,33,40);(1,3)(2,4)(5,8)(6,7)(9,12)(10,13)(11,18)(14,23)(15,19)(16,21)(17,24)(20,22)(25,29,32,28)(26,34,30,39)(27,37,35,36)(31,38,33,40);(1,4)(2,3)(5,7)(6,8)(9,13)(10,12)(11,19)(14,21)(15,18)(16,23)(17,22)(20,24)(25,30,32,26)(27,33,35,31)(28,34,29,39)(36,38,37,40);(9,11)(10,17)(12,22)(13,19)(14,20)(15,16)(18,23)(21,24)(25,31,32,33)(26,35,30,27)(28,38,29,40)(34,37,39,36)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8494368);