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 | #44996 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4333 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,13)(11,16,18,15)(12,20,22,17)(19,23,21,24);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,19)(13,16)(14,18)(17,23)(20,24)(21,22)(25,26)(27,31)(28,32)(29,30)(33,39)(34,35)(36,40)(37,38);(1,2,5,4)(3,6,8,7)(10,13)(11,18)(17,20)(19,21)(26,29)(31,34)(32,37)(39,40);(1,2,5,4)(3,6,8,7)(9,11)(10,16)(12,21)(13,15)(14,18)(17,23)(19,22)(20,24)(25,27,30,35)(26,31,29,34)(28,36,38,33)(32,40,37,39);(1,4,5,2)(3,7,8,6)(9,12)(10,17)(11,19)(13,20)(14,22)(15,23)(16,24)(18,21)(25,28,30,38)(26,32,29,37)(27,33,35,36)(31,39,34,40);(9,13,14,10)(11,15,18,16)(12,20,22,17)(19,23,21,24)(25,29,30,26)(27,34,35,31)(28,37,38,32)(33,40,36,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gh,d^2=gi,e^2=gi,f^2=i,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, 7756223);