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 | #2428 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4312 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)(25,26,32,29)(27,33,31,36)(28,35,30,34)(37,40,38,39);(1,3,5,8)(2,6,4,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,21)(22,24)(25,27,32,31)(26,33,29,36)(28,38,30,37)(34,40,35,39);(10,15)(13,23)(16,21)(17,24)(25,28,32,30)(26,34,29,35)(27,37,31,38)(33,39,36,40);(9,11)(10,16)(12,18)(13,17)(14,20)(15,21)(19,22)(23,24)(25,29,32,26)(27,36,31,33)(28,34,30,35)(37,39,38,40);(1,2,5,4)(3,6,8,7)(9,12)(10,17)(11,19)(13,21)(14,22)(15,24)(16,23)(18,20)(25,30,32,28)(26,35,29,34)(27,37,31,38)(33,39,36,40);(1,4,5,2)(3,7,8,6)(9,13,14,23)(10,18,15,19)(11,17,20,24)(12,21,22,16)(25,31,32,27)(26,36,29,33)(28,37,30,38)(34,39,35,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=h,e^2=gh,f^2=gi,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, 7686632);