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 |
| prime radical | #47708 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3987 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)(3,9)(4,6)(5,8)(7,10)(11,15)(12,13)(14,16)(17,18)(19,25)(20,27)(21,29)(22,24)(23,30)(26,28)(31,32)(34,36)(35,38)(37,45)(47,48);(2,4)(5,13)(7,9)(11,16)(17,18,22,24)(19,25,27,20)(21,23,30,29)(26,31,32,28)(33,34,39,36)(35,40,38,43)(37,41,45,44)(42,47,46,48);(3,10)(7,9)(11,16)(14,15)(17,19)(18,20)(21,26)(22,27)(23,28)(24,25)(29,31)(30,32)(33,35,39,38)(34,40,36,43)(37,42,45,46)(41,47,44,48);(1,3)(2,7)(4,9)(5,11)(6,10)(8,14)(12,15)(13,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31)(33,36,39,34)(35,40,38,43)(37,44,45,41)(42,47,46,48);(1,4,6,2)(3,9,10,7)(5,12,13,8)(11,15,16,14)(17,18)(19,20)(21,23)(22,24)(25,27)(26,28)(29,30)(31,32)(33,37)(34,41)(35,42)(36,44)(38,46)(39,45)(40,47)(43,48);(1,5,6,13)(2,8,4,12)(3,11,10,16)(7,14,9,15)(17,21,22,30)(18,23,24,29)(19,26,27,32)(20,28,25,31)(33,38,39,35)(34,40,36,43)(37,46,45,42)(41,47,44,48)Polycyclic presentation:
2^9|b^2=h,c^2=hi,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662605);