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 | #35009 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4282 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)(3,5)(4,7)(6,8)(12,20)(16,21)(18,24)(22,23)(25,26)(27,34)(28,32)(29,38)(30,33)(31,37)(35,36)(39,40);(2,3)(6,7)(9,10)(11,15)(12,21)(13,17)(14,19)(16,20)(18,22)(23,24)(25,27,30,37)(26,31,33,34)(28,35,38,40)(29,36,32,39);(1,2,5,3)(4,6,8,7)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,24)(21,23)(25,28,30,38)(26,32,33,29)(27,35,37,40)(31,39,34,36);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,18)(13,19)(14,17)(16,23)(20,24)(21,22)(25,29)(26,28)(27,36)(30,32)(31,35)(33,38)(34,40)(37,39);(1,4,5,8)(2,6,3,7)(9,12)(10,16)(11,18)(13,20)(14,22)(15,23)(17,21)(19,24)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40);(9,10)(11,14)(12,16)(13,17)(15,19)(18,22)(20,21)(23,24)(25,26)(27,31)(28,32)(29,38)(30,33)(34,37)(35,39)(36,40)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8380025);