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) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,15,19)(11,20,21,16)(12,13,17,23)(14,18,22,24)(25,26)(27,32)(28,30)(29,33)(31,35)(34,40)(36,38)(37,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,22)(13,18)(14,17)(15,21)(19,20)(23,24)(25,27)(26,31)(28,32)(29,37)(30,35)(33,40)(34,38)(36,39);(11,21)(14,22)(16,20)(18,24)(25,27,30,35)(26,31,28,32)(29,37,38,34)(33,40,36,39);(9,12)(10,13)(11,22)(14,21)(15,17)(16,18)(19,23)(20,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,40)(36,39)(37,38);(9,13)(10,17)(11,18)(12,19)(14,16)(15,23)(20,22)(21,24)(25,28,30,26)(27,32,35,31)(29,36,38,33)(34,40,37,39);(1,3,4,7)(2,5,6,8)(9,14,15,22)(10,18,19,24)(11,12,21,17)(13,16,23,20)(25,29,30,38)(26,33,28,36)(27,34,35,37)(31,39,32,40)Polycyclic presentation:
2^9|a^2=gh,c^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7720346);