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 | |
4389 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26,31,29)(27,34,30,32)(28,33,37,38)(35,40,36,39);(1,2,5,3)(4,6,8,7)(11,17)(15,21)(18,23)(22,24)(25,27,31,30)(26,32,29,34)(28,35,37,36)(33,39,38,40);(2,3)(6,7)(10,14)(15,21)(16,19)(22,24)(25,28)(26,33)(27,35)(29,38)(30,36)(31,37)(32,39)(34,40);(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,27,31,30)(26,32,29,34)(28,35,37,36)(33,39,38,40);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,29,31,26)(27,34,30,32)(28,33,37,38)(35,39,36,40);(1,4)(2,6)(3,7)(5,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,30,31,27)(26,34,29,32)(28,36,37,35)(33,40,38,39)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8001905);