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 | |
4243 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)(3,7)(4,12)(5,13)(6,8)(9,14)(10,16)(11,15)(18,21)(19,25)(24,29)(26,31)(34,38)(35,41)(36,44)(46,48);(1,3,6,9)(2,7,8,14)(4,10,13,15)(5,11,12,16)(17,18,22,21)(19,23,25,27)(20,24,28,29)(26,30,31,32)(33,34,37,38)(35,39,41,42)(36,40,44,45)(43,46,47,48);(2,8)(3,9)(4,13)(11,16)(19,25)(20,28)(23,27)(24,29)(33,34,37,38)(35,42,41,39)(36,40,44,45)(43,48,47,46);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,4)(2,5)(3,10)(6,13)(7,11)(8,12)(9,15)(14,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,47)(45,48);(1,5)(2,4)(3,11)(6,12)(7,10)(8,13)(9,16)(14,15)(17,21,22,18)(19,27,25,23)(20,29,28,24)(26,32,31,30)(33,36,37,44)(34,40,38,45)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^9|b^2=gi,c^2=hi,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, 8034432);