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 | |
3896 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,4)(3,9)(8,12)(10,15)(17,18,23,25)(19,28,29,24)(20,30,27,22)(21,26,31,32)(33,34,39,36)(35,42,38,40)(37,44,46,41)(43,47,45,48);(1,2,6,4)(3,7,9,11)(5,12,13,8)(10,16,15,14)(17,19)(18,24)(20,21)(22,26)(23,29)(25,28)(27,31)(30,32)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(3,9)(7,11)(10,15)(14,16)(18,25)(19,29)(20,27)(26,32)(33,36,39,34)(35,40,38,42)(37,44,46,41)(43,47,45,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,20,23,27)(18,22,25,30)(19,21,29,31)(24,26,28,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,4,6,2)(3,7,9,11)(5,12,13,8)(10,14,15,16)(17,21)(18,26)(19,27)(20,29)(22,28)(23,31)(24,30)(25,32)(33,35,39,38)(34,42,36,40)(37,43,46,45)(41,48,44,47);(1,5,6,13)(2,8,4,12)(3,10,9,15)(7,14,11,16)(17,22)(18,27)(19,26)(20,25)(21,28)(23,30)(24,31)(29,32)(33,38,39,35)(34,40,36,42)(37,45,46,43)(41,47,44,48)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=hi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9297620);