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 | |
4079 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,7)(3,8,11,6)(4,12,13,9)(10,14,16,15)(17,18,22,26)(19,23,27,30)(20,24,29,32)(21,25,31,28)(33,34,39,44)(35,40,37,42)(36,46,45,41)(38,43,47,48);(1,3,5,11)(2,6,7,8)(4,10,13,16)(9,14,12,15)(17,19)(18,23)(20,28)(21,24)(22,27)(25,29)(26,30)(31,32)(33,35,39,37)(34,40,44,42)(36,47,45,38)(41,43,46,48);(2,7)(4,13)(6,8)(10,16)(19,27)(20,29)(23,30)(24,32)(33,36,39,45)(34,41,44,46)(35,38,37,47)(40,48,42,43);(1,2)(3,6)(4,9)(5,7)(8,11)(10,14)(12,13)(15,16)(17,20)(18,24)(19,28)(21,23)(22,29)(25,27)(26,32)(30,31)(33,37,39,35)(34,42,44,40)(36,38,45,47)(41,48,46,43);(1,3)(2,8)(4,10)(5,11)(6,7)(9,15)(12,14)(13,16)(17,21)(18,25)(19,24)(20,30)(22,31)(23,29)(26,28)(27,32)(33,38)(34,43)(35,45)(36,37)(39,47)(40,41)(42,46)(44,48);(1,4,5,13)(2,9,7,12)(3,10,11,16)(6,14,8,15)(17,19)(18,23)(20,28)(21,24)(22,27)(25,29)(26,30)(31,32)(33,34)(35,40)(36,46)(37,42)(38,43)(39,44)(41,45)(47,48)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8812465);