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 | |
4224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,4,11,7)(5,8,13,15)(10,12,16,14)(17,18,22,25)(19,23,26,30)(20,21,24,29)(27,28,31,32)(33,34)(35,46)(36,43)(37,42)(38,39)(40,45)(41,47)(44,48);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,35,40,47)(34,41,45,46)(36,38,37,48)(39,43,44,42);(5,13)(8,15)(10,16)(12,14)(19,26)(23,30)(27,31)(28,32)(33,36,40,37)(34,42,45,43)(35,38,47,48)(39,46,44,41);(1,4,6,7)(2,3,9,11)(5,12,13,14)(8,10,15,16)(17,20)(18,21)(19,27)(22,24)(23,28)(25,29)(26,31)(30,32)(33,37,40,36)(34,43,45,42)(35,38,47,48)(39,46,44,41);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,16)(14,15)(17,21)(18,24)(19,28)(20,25)(22,29)(23,31)(26,32)(27,30)(33,38)(34,44)(35,37)(36,47)(39,45)(40,48)(41,42)(43,46);(1,5,6,13)(2,8,9,15)(3,10,11,16)(4,12,7,14)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,39,40,44)(34,38,45,48)(35,43,47,42)(36,41,37,46)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=h,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, 8184014);