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 | #52013 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10,14,16)(11,17,18,12)(13,21,22,15)(19,20,24,23)(25,26)(27,32)(28,37)(29,39)(30,33)(31,34)(35,40)(36,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,22)(23,24)(27,34)(31,32)(35,38)(36,40);(2,5)(6,8)(11,18)(12,17)(19,24)(20,23)(25,27,30,34)(26,31,33,32)(28,38,39,35)(29,36,37,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,19)(14,18)(15,20)(16,17)(21,23)(22,24)(25,27)(26,32)(28,35)(29,36)(30,34)(31,33)(37,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,24)(22,23)(25,28)(26,29)(27,35)(30,39)(31,40)(32,36)(33,37)(34,38);(1,3)(2,6)(4,7)(5,8)(9,13,14,22)(10,15,16,21)(11,19,18,24)(12,20,17,23)(25,29)(26,28)(27,36)(30,37)(31,38)(32,35)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=g,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8032780);