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 | #2670 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3995 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,11)(4,15)(5,10)(6,7)(8,14)(9,13)(12,16)(33,34)(35,44)(36,38)(37,41)(39,43)(40,45)(42,48)(46,47);(1,3,7,14)(2,8,6,11)(4,12,13,5)(9,16,15,10)(17,18,22,26)(19,24,29,23)(20,21,25,32)(27,31,30,28)(35,45)(37,46)(40,44)(41,47);(2,6)(4,13)(5,12)(8,11)(17,19,22,29)(18,23,26,24)(20,30,25,27)(21,31,32,28)(33,35,39,45)(34,40,43,44)(36,37,42,46)(38,47,48,41);(1,4)(2,9)(3,12)(5,14)(6,15)(7,13)(8,16)(10,11)(17,20,22,25)(18,21,26,32)(19,27,29,30)(23,28,24,31)(33,36,39,42)(34,38,43,48)(35,46,45,37)(40,41,44,47);(1,5,7,12)(2,10,6,16)(3,13,14,4)(8,15,11,9)(17,19)(18,24)(20,30)(21,28)(22,29)(23,26)(25,27)(31,32)(33,37,39,46)(34,41,43,47)(35,36,45,42)(38,40,48,44);(1,6,7,2)(3,11,14,8)(4,15,13,9)(5,16,12,10)(17,21)(18,25)(19,28)(20,26)(22,32)(23,30)(24,27)(29,31)(33,38)(34,42)(35,41)(36,43)(37,40)(39,48)(44,46)(45,47)Polycyclic presentation:
2^9|b^2=hi,c^2=i,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8901452);