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 | |
4229 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)(3,8)(4,6)(5,9)(7,11)(10,15)(12,13)(14,16)(17,18,22,25)(19,23,26,30)(20,21,24,29)(27,28,31,32)(33,34,40,44)(35,45,46,39)(36,42,37,41)(38,48,47,43);(1,3)(2,7)(4,8)(5,10)(6,11)(9,14)(12,15)(13,16)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,35,40,46)(34,39,44,45)(36,38,37,47)(41,48,42,43);(5,13)(9,12)(10,16)(14,15)(19,26)(23,30)(27,31)(28,32)(33,36,40,37)(34,41,44,42)(35,38,46,47)(39,48,45,43);(1,4,6,2)(3,8,11,7)(5,12,13,9)(10,15,16,14)(17,20)(18,21)(19,27)(22,24)(23,28)(25,29)(26,31)(30,32)(33,37,40,36)(34,42,44,41)(35,38,46,47)(39,48,45,43);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,21)(18,24)(19,28)(20,25)(22,29)(23,31)(26,32)(27,30)(33,38)(34,43)(35,37)(36,46)(39,41)(40,47)(42,45)(44,48);(1,5,6,13)(2,9,4,12)(3,10,11,16)(7,14,8,15)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,39,40,45)(34,35,44,46)(36,43,37,48)(38,41,47,42)Polycyclic presentation:
2^9|a^2=hi,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, 8184013);