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 | |
4128 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(19,25)(23,26)(27,31)(30,32)(33,34,39,45)(35,40,44,38)(36,43,47,41)(37,48,46,42);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35,39,44)(34,40,45,38)(36,37,47,46)(41,42,43,48);(2,3)(7,9)(8,10)(14,15)(18,22)(23,26)(24,28)(30,32)(33,36,39,47)(34,41,45,43)(35,46,44,37)(38,42,40,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32)(33,37,39,46)(34,42,45,48)(35,36,44,47)(38,43,40,41);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31)(33,36)(34,43)(35,46)(37,44)(38,48)(39,47)(40,42)(41,45);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,38,39,40)(34,44,45,35)(36,42,47,48)(37,43,46,41)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9945679);