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 |
| prime radical | #53062 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4187 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(10,13)(11,17)(16,21)(18,23)(25,26,31,32)(27,28,38,33)(29,34,30,35)(36,39,37,40);(1,2,5,3)(4,6,8,7)(9,10,14,13)(11,15,17,19)(12,16,20,21)(18,22,23,24)(26,32)(28,33)(34,35)(39,40);(2,3)(6,7)(12,20)(16,21)(18,23)(22,24)(25,27)(26,33)(28,32)(29,36)(30,37)(31,38)(34,40)(35,39);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,28,31,33)(26,27,32,38)(29,39,30,40)(34,36,35,37);(1,3,5,2)(4,7,8,6)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,29,31,30)(26,34,32,35)(27,36,38,37)(28,39,33,40);(1,4)(2,6)(3,7)(5,8)(9,13,14,10)(11,19,17,15)(12,21,20,16)(18,24,23,22)(25,30,31,29)(26,35,32,34)(27,37,38,36)(28,40,33,39)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=h,e^2=hi,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, 8002079);