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 | |
4464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(25,26,30,29)(27,31,33,35)(28,32,36,37)(34,38,39,40);(1,2)(3,7)(4,8)(5,6)(9,10,14,15)(11,12,16,20)(13,17,21,22)(18,19,23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(1,3,5,4)(2,7,6,8)(10,15)(11,16)(13,21)(19,24)(27,33)(31,35)(34,39)(38,40);(1,3,5,4)(2,7,6,8)(9,11)(10,12)(13,18)(14,16)(15,20)(17,19)(21,23)(22,24)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(1,3,5,4)(2,7,6,8)(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,29,30,26)(27,35,33,31)(28,32,36,37)(34,38,39,40);(1,4,5,3)(2,8,6,7)(9,13)(10,17)(11,18)(12,19)(14,21)(15,22)(16,23)(20,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=gh,d^2=gh,e^2=gi,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7681122);