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 | |
4052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,10,4,7)(5,13,14,8)(11,15,12,16)(17,18,23,28)(19,29,21,24)(20,26,31,25)(22,30,32,27)(33,34)(35,42)(36,40)(37,41)(38,43)(39,46)(44,47)(45,48);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,19,23,21)(18,24,28,29)(20,27,31,30)(22,26,32,25)(33,35)(34,39)(36,44)(37,47)(38,46)(40,45)(41,48)(42,43);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,20)(18,25)(19,27)(21,30)(22,24)(23,31)(26,28)(29,32)(33,36,38,41)(34,40,43,37)(35,44,46,48)(39,45,42,47);(1,2)(3,7)(4,10)(5,8)(6,9)(11,15)(12,16)(13,14)(17,20)(18,26)(19,30)(21,27)(22,24)(23,31)(25,28)(29,32)(33,37,38,40)(34,41,43,36)(35,45,46,47)(39,44,42,48);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,21,23,19)(18,24,28,29)(20,30,31,27)(22,26,32,25)(33,34)(35,39)(36,40)(37,41)(38,43)(42,46)(44,45)(47,48);(1,5,6,14)(2,8,9,13)(3,11,4,12)(7,15,10,16)(17,22,23,32)(18,27,28,30)(19,26,21,25)(20,24,31,29)(33,35)(34,42)(36,44)(37,45)(38,46)(39,43)(40,47)(41,48)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10362866);