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 | #43097 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4288 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)(3,7)(4,5)(6,8)(9,10,14,16)(11,17,18,12)(13,15,21,22)(19,23,24,20);(3,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,22)(23,24)(25,26,30,29)(27,31,33,35)(28,32,36,38)(34,39,37,40);(1,3,5,8)(2,6,4,7)(11,18)(12,17)(13,21)(15,22)(26,29)(27,33)(28,36)(39,40);(3,8)(6,7)(9,11,14,18)(10,12,16,17)(13,19,21,24)(15,20,22,23)(25,27)(26,31)(28,37)(29,35)(30,33)(32,40)(34,36)(38,39);(1,3)(2,7)(4,6)(5,8)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,28)(26,32)(27,34)(29,38)(30,36)(31,39)(33,37)(35,40);(1,4,5,2)(3,6,8,7)(9,13,14,21)(10,15,16,22)(11,19,18,24)(12,20,17,23)(25,29,30,26)(27,35,33,31)(28,38,36,32)(34,40,37,39)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=h,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8929109);