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 | |
4208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,4,8,7)(5,13,14,9)(11,15,16,12)(17,18,22,24)(19,26,21,23)(20,28,30,25)(27,31,29,32)(33,34)(35,40)(36,38)(37,44)(39,42)(41,45)(43,46)(47,48);(1,2)(3,7)(4,8)(5,9)(6,10)(11,15)(12,16)(13,14)(17,19,22,21)(18,23,24,26)(20,29,30,27)(25,32,28,31)(35,42)(39,40)(43,48)(46,47);(3,8)(4,7)(11,16)(12,15)(18,24)(20,30)(23,26)(27,29)(33,35,38,42)(34,39,36,40)(37,43,45,48)(41,46,44,47);(1,3,6,8)(2,7,10,4)(5,11,14,16)(9,15,13,12)(17,20,22,30)(18,25,24,28)(19,27,21,29)(23,31,26,32)(33,35)(34,40)(36,39)(37,43)(38,42)(41,47)(44,46)(45,48);(1,4,6,7)(2,8,10,3)(5,12,14,15)(9,16,13,11)(17,18)(19,23)(20,25)(21,26)(22,24)(27,31)(28,30)(29,32)(33,36,38,34)(35,39,42,40)(37,44,45,41)(43,46,48,47);(1,5,6,14)(2,9,10,13)(3,11,8,16)(4,12,7,15)(17,21,22,19)(18,26,24,23)(20,29,30,27)(25,32,28,31)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=i,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034615);