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 | |
4088 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)(3,12)(4,14)(5,6)(7,9)(8,13)(10,16)(11,15)(17,18,23,26)(19,24,27,22)(20,21,25,31)(28,29,32,30)(34,39)(36,43)(41,45)(44,48);(1,3,7,13)(2,8,9,12)(4,15,16,5)(6,10,11,14)(17,19,23,27)(18,24,26,22)(20,28,25,32)(21,29,31,30)(34,39)(35,40)(41,45)(42,47);(2,9)(6,11)(8,12)(10,14)(19,27)(22,24)(28,32)(29,30)(33,34,38,39)(35,36,40,43)(37,45,46,41)(42,48,47,44);(1,4,7,16)(2,10,9,14)(3,5,13,15)(6,12,11,8)(17,20)(18,21)(19,28)(22,30)(23,25)(24,29)(26,31)(27,32)(33,35)(34,36)(37,42)(38,40)(39,43)(41,44)(45,48)(46,47);(1,5)(2,11)(3,4)(6,9)(7,15)(8,14)(10,12)(13,16)(17,21)(18,25)(19,29)(20,26)(22,28)(23,31)(24,32)(27,30)(33,36)(34,40)(35,39)(37,44)(38,43)(41,47)(42,45)(46,48);(1,6,7,11)(2,5,9,15)(3,10,13,14)(4,12,16,8)(17,22)(18,19)(20,30)(21,28)(23,24)(25,29)(26,27)(31,32)(33,37)(34,41)(35,42)(36,44)(38,46)(39,45)(40,47)(43,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8191690);