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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,8,6,16)(4,5,10,9)(12,14,13,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,29)(28,31)(30,32)(33,34,38,37)(35,39,44,42)(36,40,45,46)(41,47,43,48);(1,3,7,6)(2,8,11,16)(4,13,10,12)(5,15,9,14)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,32)(30,31)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47);(1,4,7,10)(2,9,11,5)(3,12,6,13)(8,15,16,14)(17,20,22,26)(18,23,25,19)(21,29,30,28)(24,27,32,31)(33,36,38,45)(34,40,37,46)(35,41,44,43)(39,47,42,48);(1,4)(2,5)(3,13)(6,12)(7,10)(8,15)(9,11)(14,16)(17,20)(18,19)(21,29)(22,26)(23,25)(24,31)(27,32)(28,30)(33,37,38,34)(35,42,44,39)(36,40,45,46)(41,47,43,48);(1,5)(2,10)(3,14)(4,11)(6,15)(7,9)(8,13)(12,16)(17,21,22,30)(18,24,25,32)(19,27,23,31)(20,28,26,29)(33,36)(34,40)(35,43)(37,46)(38,45)(39,48)(41,44)(42,47);(1,6,7,3)(2,8,11,16)(4,12,10,13)(5,15,9,14)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gi,d^2=hi,e^2=g,f^2=h,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, 8900852);