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 | #2431 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4237 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(25,26)(27,35)(28,30)(29,33)(31,32)(34,38)(36,37)(39,40);(1,3,5,8)(2,6,7,4)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,21)(22,24)(26,32)(28,38)(33,35)(36,39);(1,3)(2,4)(5,8)(6,7)(10,15)(13,23)(16,21)(17,24)(25,27,31,29)(26,33,32,35)(28,36,38,39)(30,40,34,37);(1,4)(2,3)(5,6)(7,8)(9,11)(10,16)(12,18)(13,17)(14,20)(15,21)(19,22)(23,24)(25,28)(26,34)(27,36)(29,39)(30,32)(31,38)(33,37)(35,40);(9,12)(10,17)(11,19)(13,21)(14,22)(15,24)(16,23)(18,20)(25,29,31,27)(26,35,32,33)(28,36,38,39)(30,40,34,37);(9,13,14,23)(10,18,15,19)(11,17,20,24)(12,21,22,16)(25,30,31,34)(26,28,32,38)(27,37,29,40)(33,39,35,36)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8229581);