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 | |
4216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,3)(7,9)(8,10)(14,15)(17,18,23,26)(19,24,27,22)(20,21,25,31)(28,29,32,30)(33,34,40,39)(35,44,38,41)(36,43,37,42)(45,47,46,48);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19,23,27)(18,24,26,22)(20,28,25,32)(21,29,31,30)(33,35,40,38)(34,41,39,44)(36,46,37,45)(42,48,43,47);(5,13)(8,10)(11,16)(14,15)(19,27)(22,24)(28,32)(29,30)(33,36,40,37)(34,42,39,43)(35,45,38,46)(41,47,44,48);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,20)(18,21)(19,28)(22,30)(23,25)(24,29)(26,31)(27,32)(33,37,40,36)(34,43,39,42)(35,46,38,45)(41,48,44,47);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21)(18,25)(19,29)(20,26)(22,28)(23,31)(24,32)(27,30)(33,38,40,35)(34,44,39,41)(36,46,37,45)(42,48,43,47);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22)(18,19)(20,30)(21,28)(23,24)(25,29)(26,27)(31,32)(33,39,40,34)(35,44,38,41)(36,43,37,42)(45,48,46,47)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=h,d^2=g,e^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, 8191647);