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 | |
4149 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(7,9)(8,13)(14,15)(17,18,22,25)(19,23,26,30)(20,21,24,29)(27,28,31,32)(33,34,39,44)(35,42,37,40)(36,45,47,41)(38,43,48,46);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,35,39,37)(34,40,44,42)(36,46,47,43)(38,45,48,41);(4,12)(8,13)(10,16)(14,15)(19,26)(23,30)(27,31)(28,32)(33,35,39,37)(34,40,44,42)(36,43,47,46)(38,41,48,45);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,20)(18,21)(19,27)(22,24)(23,28)(25,29)(26,31)(30,32)(33,36)(34,41)(35,43)(37,46)(38,42)(39,47)(40,48)(44,45);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,21)(18,24)(19,28)(20,25)(22,29)(23,31)(26,32)(27,30)(33,37,39,35)(34,42,44,40)(36,43,47,46)(38,41,48,45);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,38,39,48)(34,43,44,46)(35,45,37,41)(36,42,47,40)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,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, 8184021);