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 | |
4174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(7,10)(8,13)(14,16)(17,18,23,25)(19,28,29,24)(20,30,27,22)(21,26,31,32);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,21)(22,26)(23,29)(25,28)(27,31)(30,32)(33,34,38,39)(35,37,41,46)(36,40,43,47)(42,45,44,48);(3,9)(4,12)(7,10)(8,13)(18,25)(19,29)(20,27)(26,32)(34,39)(36,43)(37,46)(42,44);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,20,23,27)(18,22,25,30)(19,21,29,31)(24,26,28,32)(33,35,38,41)(34,37,39,46)(36,44,43,42)(40,48,47,45);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,15)(13,16)(17,21)(18,26)(19,27)(20,29)(22,28)(23,31)(24,30)(25,32)(33,36)(34,40)(35,42)(37,45)(38,43)(39,47)(41,44)(46,48);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,22)(18,27)(19,26)(20,25)(21,28)(23,30)(24,31)(29,32)(33,37)(34,41)(35,39)(36,45)(38,46)(40,44)(42,47)(43,48)Polycyclic presentation:
2^9|a^2=h,b^2=hi,d^2=i,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, 9416118);