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 | |
4166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,8,12,15)(4,5,9,14)(10,16,13,11)(17,18)(19,22)(20,29)(21,25)(23,26)(24,28)(27,32)(30,31)(33,34)(35,38)(36,44)(37,41)(39,43)(40,45)(42,47)(46,48);(2,7)(4,9)(8,15)(10,13)(17,19)(18,22)(20,21)(23,27)(24,30)(25,29)(26,32)(28,31)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,47)(45,48);(4,9)(5,14)(10,13)(11,16)(19,27)(21,31)(22,32)(25,30)(33,34,37,41)(35,39,43,38)(36,44,45,40)(42,46,48,47);(1,3,6,12)(2,8,7,15)(4,13,9,10)(5,11,14,16)(17,20)(18,24)(19,21)(22,30)(23,28)(25,32)(26,29)(27,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,4,6,9)(2,5,7,14)(3,10,12,13)(8,16,15,11)(17,21)(18,25)(19,28)(20,27)(22,24)(23,31)(26,30)(29,32)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,45)(47,48);(1,5)(2,9)(3,11)(4,7)(6,14)(8,10)(12,16)(13,15)(17,22)(18,19)(20,30)(21,24)(23,32)(25,28)(26,27)(29,31)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,44)(43,48)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8941677);