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 | |
3969 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)(3,4)(5,13)(6,7)(8,15)(9,10)(11,14)(12,16)(17,18,23,26)(19,20,21,24)(22,30,31,25)(27,32,29,28)(33,34,40,45)(35,39,44,41)(36,42,37,46)(38,43,47,48);(1,3)(2,4)(5,11)(6,9)(7,10)(8,12)(13,14)(15,16)(17,19)(18,24)(20,26)(21,23)(22,27)(25,32)(28,30)(29,31)(33,35,40,44)(34,41,45,39)(36,38,37,47)(42,48,46,43);(2,7)(3,9)(8,13)(11,16)(17,20,23,24)(18,19,26,21)(22,28,31,32)(25,27,30,29)(33,36,40,37)(34,42,45,46)(35,38,44,47)(39,43,41,48);(1,2)(3,10)(4,9)(5,8)(6,7)(11,14)(12,16)(13,15)(17,20)(18,19)(21,26)(22,28)(23,24)(25,27)(29,30)(31,32)(33,37,40,36)(34,42,45,46)(35,38,44,47)(39,48,41,43);(1,4,6,10)(2,3,7,9)(5,14,15,12)(8,16,13,11)(17,21,23,19)(18,24,26,20)(22,27,31,29)(25,28,30,32)(33,38)(34,43)(35,37)(36,44)(39,46)(40,47)(41,42)(45,48);(1,5)(2,8)(3,11)(4,12)(6,15)(7,13)(9,16)(10,14)(17,22,23,31)(18,25,26,30)(19,27,21,29)(20,28,24,32)(33,39,40,41)(34,44,45,35)(36,43,37,48)(38,42,47,46)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=g,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154357);