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 | |
4037 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18,21,25)(19,22,27,23)(20,28,30,24)(26,32,29,31)(34,36)(37,43)(41,44)(45,48);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,16)(14,15)(17,19)(18,22)(20,29)(21,27)(23,25)(24,32)(26,30)(28,31)(33,34,39,36)(35,37,40,43)(38,44,46,41)(42,48,47,45);(2,6)(7,10)(8,12)(14,16)(17,19,21,27)(18,23,25,22)(20,26,30,29)(24,32,28,31)(33,35,39,40)(34,37,36,43)(38,42,46,47)(41,45,44,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19)(18,22)(20,26)(21,27)(23,25)(24,31)(28,32)(29,30)(33,36,39,34)(35,37,40,43)(38,44,46,41)(42,45,47,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,18)(19,23)(20,24)(21,25)(22,27)(26,32)(28,30)(29,31)(33,37)(34,40)(35,36)(38,45)(39,43)(41,47)(42,44)(46,48);(1,4,5,13)(2,8,6,12)(3,11,9,15)(7,14,10,16)(17,20)(18,24)(19,26)(21,30)(22,31)(23,32)(25,28)(27,29)(33,38)(34,41)(35,42)(36,44)(37,45)(39,46)(40,47)(43,48)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,d^2=g,f^2=hi,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=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8784347);