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 | |
4169 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,11,5)(4,6,9,15)(12,13,16,14)(17,18)(19,24)(20,29)(21,25)(22,28)(23,27)(26,32)(30,31)(34,39)(35,41)(36,43)(45,48);(1,3,7,11)(2,8,10,5)(4,12,9,16)(6,13,15,14)(17,19,23,30)(18,24,27,31)(20,26,21,28)(22,29,32,25)(33,34,38,39)(35,37,41,46)(36,40,43,47)(42,45,44,48);(3,11)(5,8)(12,16)(13,14)(17,20,23,21)(18,25,27,29)(19,28,30,26)(22,24,32,31)(33,35,38,41)(34,37,39,46)(36,44,43,42)(40,48,47,45);(1,4)(2,6)(3,12)(5,14)(7,9)(8,13)(10,15)(11,16)(17,21,23,20)(18,25,27,29)(19,26,30,28)(22,24,32,31)(33,34)(35,37)(36,40)(38,39)(41,46)(42,45)(43,47)(44,48);(1,5)(2,3)(4,14)(6,12)(7,8)(9,13)(10,11)(15,16)(17,22)(18,26)(19,29)(20,31)(21,24)(23,32)(25,30)(27,28)(33,36,38,43)(34,40,39,47)(35,42,41,44)(37,45,46,48);(1,6)(2,9)(3,13)(4,10)(5,12)(7,15)(8,16)(11,14)(17,18)(19,24)(20,25)(21,29)(22,26)(23,27)(28,32)(30,31)(33,37)(34,41)(35,39)(36,45)(38,46)(40,44)(42,47)(43,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=h,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8429360);