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 | |
4176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,10,7)(4,8,13,14)(11,16,12,15)(17,18)(19,24)(20,26)(21,30)(22,28)(23,25)(27,29)(31,32)(33,34,38,41)(35,36,43,39)(37,40,46,48)(42,44,45,47);(2,6)(3,10)(8,14)(11,12)(17,19)(18,24)(20,29)(21,31)(22,25)(23,28)(26,27)(30,32)(33,35)(34,39)(36,41)(37,45)(38,43)(40,44)(42,46)(47,48);(2,6)(7,9)(8,14)(15,16)(18,25)(19,28)(21,29)(26,32)(37,46)(40,48)(42,45)(44,47);(1,3,5,10)(2,7,6,9)(4,12,13,11)(8,16,14,15)(17,20)(18,26)(19,21)(22,27)(23,31)(24,30)(25,32)(28,29)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,48,45);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,21)(18,27)(19,20)(22,26)(23,29)(24,32)(25,30)(28,31)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,48)(43,45);(1,4,5,13)(2,8,6,14)(3,11,10,12)(7,15,9,16)(17,22)(18,28)(19,25)(20,30)(21,26)(23,24)(27,31)(29,32)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=g,d^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8540690);