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 | |
3775 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)(3,6)(4,13)(5,9)(7,10)(8,12)(11,16)(14,15)(18,23)(21,29)(24,30)(28,32)(33,34,40,44)(35,45,47,38)(36,46,48,41)(37,43,39,42);(1,3)(2,6)(4,5)(7,11)(8,14)(9,13)(10,16)(12,15)(20,27)(24,30)(26,31)(28,32)(33,35,40,47)(34,38,44,45)(36,43,48,42)(37,41,39,46);(3,11)(4,12)(6,16)(8,13)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,36,40,48)(34,41,44,46)(35,42,47,43)(37,45,39,38);(1,4)(2,8)(3,5)(6,14)(7,12)(9,16)(10,13)(11,15)(17,19,22,25)(18,21,23,29)(20,26,27,31)(24,28,30,32)(33,37,40,39)(34,42,44,43)(35,41,47,46)(36,45,48,38);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,14)(13,16)(17,20,22,27)(18,24,23,30)(19,26,25,31)(21,28,29,32)(33,38)(34,35)(36,37)(39,48)(40,45)(41,42)(43,46)(44,47);(1,6)(2,3)(4,14)(5,8)(7,16)(9,12)(10,11)(13,15)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,39,40,37)(34,43,44,42)(35,46,47,41)(36,38,48,45)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8835239);