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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3841 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)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(19,25)(23,26)(27,31)(30,32)(33,34,39,37)(35,40,38,46)(36,43,45,41)(42,48,44,47);(2,7)(5,8)(9,12)(13,15)(17,18)(19,26)(20,28)(21,22)(23,25)(24,29)(27,30)(31,32)(33,35,39,38)(34,40,37,46)(36,42,45,44)(41,47,43,48);(2,7)(3,10)(9,12)(11,16)(18,22)(23,26)(24,28)(30,32)(33,36)(34,41)(35,42)(37,43)(38,44)(39,45)(40,47)(46,48);(1,3,6,10)(2,8,7,5)(4,11,14,16)(9,15,12,13)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32);(1,4)(2,9)(3,11)(5,13)(6,14)(7,12)(8,15)(10,16)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32)(33,37,39,34)(35,40,38,46)(36,43,45,41)(42,47,44,48);(1,5,6,8)(2,10,7,3)(4,13,14,15)(9,16,12,11)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,38,39,35)(34,40,37,46)(36,44,45,42)(41,47,43,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=g,e^2=ghi,f^2=h,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, 8663290);