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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,22,27)(19,25,30,23)(20,31,29,24)(21,32,28,26)(33,34)(35,42)(36,37)(38,40)(39,43)(41,48)(44,45)(46,47);(1,2,7,9)(3,10,4,5)(6,14,15,8)(11,13,12,16)(17,19,22,30)(18,23,27,25)(20,28,29,21)(24,26,31,32)(33,35,39,38)(34,40,43,42)(36,46,41,44)(37,45,48,47);(1,3,7,4)(2,5,9,10)(6,11,15,12)(8,13,14,16)(17,18)(19,23)(20,31)(21,26)(22,27)(24,29)(25,30)(28,32)(33,36,39,41)(34,37,43,48)(35,44,38,46)(40,47,42,45);(1,4,7,3)(2,5,9,10)(6,12,15,11)(8,13,14,16)(17,20,22,29)(18,24,27,31)(19,28,30,21)(23,26,25,32)(33,37)(34,41)(35,45)(36,43)(38,47)(39,48)(40,44)(42,46);(1,5)(2,3)(4,9)(6,13)(7,10)(8,11)(12,14)(15,16)(17,19)(18,25)(20,28)(21,29)(22,30)(23,27)(24,32)(26,31)(33,35,39,38)(34,42,43,40)(36,44,41,46)(37,45,48,47);(1,6)(2,8)(3,11)(4,12)(5,13)(7,15)(9,14)(10,16)(17,21,22,28)(18,26,27,32)(19,29,30,20)(23,24,25,31)(33,38,39,35)(34,40,43,42)(36,46,41,44)(37,47,48,45)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=i,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9174226);