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 | |
4103 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,11)(4,9)(5,13)(6,7)(8,12)(10,16)(14,15)(18,23)(19,26)(25,30)(27,32)(33,34,39,43)(35,38,37,41)(36,42,47,40)(44,46,45,48);(2,7)(4,15)(8,11)(10,13)(17,18,22,23)(19,24,26,20)(21,25,29,30)(27,31,32,28)(33,35,39,37)(34,38,43,41)(36,45,47,44)(40,46,42,48);(2,7)(3,12)(4,15)(5,16)(19,26)(20,24)(21,29)(25,30)(33,36,39,47)(34,40,43,42)(35,44,37,45)(38,48,41,46);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,19,22,26)(18,24,23,20)(21,27,29,32)(25,31,30,28)(33,37,39,35)(34,41,43,38)(36,44,47,45)(40,48,42,46);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,20,22,24)(18,19,23,26)(21,28,29,31)(25,27,30,32)(33,36)(34,42)(35,44)(37,45)(38,46)(39,47)(40,43)(41,48);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,38,39,41)(34,35,43,37)(36,46,47,48)(40,45,42,44)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=i,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8900346);