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 | |
3991 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,22,23)(19,27,30,24)(20,25,31,29)(21,28,32,26)(33,34)(35,43)(36,41)(37,42)(38,39)(40,44)(45,48)(46,47);(1,2,6,7)(3,4,8,11)(5,9,13,14)(10,12,15,16)(18,23)(20,31)(24,27)(26,28)(35,44)(37,45)(40,43)(42,48);(2,7)(4,11)(5,13)(10,15)(17,19,22,30)(18,24,23,27)(20,26,31,28)(21,25,32,29)(33,35,39,44)(34,40,38,43)(36,37,47,45)(41,48,46,42);(1,3)(2,4)(5,10)(6,8)(7,11)(9,12)(13,15)(14,16)(17,19)(18,24)(20,26)(21,29)(22,30)(23,27)(25,32)(28,31)(33,36,39,47)(34,41,38,46)(35,45,44,37)(40,42,43,48);(1,4)(2,8)(3,7)(5,12)(6,11)(9,15)(10,14)(13,16)(17,20)(18,25)(19,28)(21,24)(22,31)(23,29)(26,30)(27,32)(33,37,39,45)(34,42,38,48)(35,36,44,47)(40,46,43,41);(1,5)(2,9)(3,10)(4,12)(6,13)(7,14)(8,15)(11,16)(17,21)(18,26)(19,29)(20,24)(22,32)(23,28)(25,30)(27,31)(33,38,39,34)(35,40,44,43)(36,46,47,41)(37,48,45,42)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=h,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9397329);