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 | |
4066 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)(4,6)(5,8)(7,10)(9,14)(12,16)(13,15)(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);(1,3,7,5)(2,8,10,11)(4,12,14,15)(6,13,9,16)(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);(4,14)(6,9)(12,15)(13,16)(19,27)(20,29)(23,30)(24,32)(33,36,39,45)(34,41,44,46)(35,38,37,47)(40,48,42,43);(1,4)(2,9)(3,12)(5,15)(6,10)(7,14)(8,16)(11,13)(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);(1,5,7,3)(2,8,10,11)(4,12,14,15)(6,16,9,13)(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);(1,6,7,9)(2,4,10,14)(3,13,5,16)(8,12,11,15)(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=i,b^2=gh,c^2=h,d^2=h,e^2=g,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, 8812470);