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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4219 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,8)(9,14)(10,15)(17,18)(19,28)(20,30)(21,22)(23,25)(24,29)(26,32)(27,31)(33,34,38,39)(35,37,41,47)(36,40,45,48)(42,46,43,44);(1,2,6,7)(3,4,8,12)(5,9,13,14)(10,11,15,16)(17,19,23,29)(18,24,25,28)(20,31,32,21)(22,26,27,30)(33,35)(34,37)(36,43)(38,41)(39,47)(40,44)(42,45)(46,48);(5,13)(9,14)(10,15)(11,16)(18,25)(22,27)(24,28)(26,30)(34,39)(35,41)(40,48)(42,43);(1,3)(2,4)(5,10)(6,8)(7,12)(9,11)(13,15)(14,16)(17,20,23,32)(18,26,25,30)(19,21,29,31)(22,28,27,24)(33,36,38,45)(34,40,39,48)(35,42,41,43)(37,46,47,44);(1,4)(2,8)(3,7)(5,11)(6,12)(9,15)(10,14)(13,16)(17,21)(18,27)(19,20)(22,25)(23,31)(24,30)(26,28)(29,32)(33,37)(34,35)(36,44)(38,47)(39,41)(40,43)(42,48)(45,46);(1,5)(2,9)(3,10)(4,11)(6,13)(7,14)(8,15)(12,16)(17,22,23,27)(18,21,25,31)(19,26,29,30)(20,28,32,24)(33,35)(34,37)(36,42)(38,41)(39,47)(40,46)(43,45)(44,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=h,f^2=hi,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, 8231471);