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 | |
4366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,12,8)(4,9,14,13)(5,15,16,10)(18,19)(23,25)(24,26)(30,31)(33,34,38,39)(35,41,42,36)(37,45,46,40)(43,44,48,47);(2,7)(4,14)(8,11)(10,15)(17,18,22,19)(20,23,27,25)(21,24,29,26)(28,30,32,31)(33,35,38,42)(34,36,39,41)(37,43,46,48)(40,44,45,47);(2,7)(3,12)(4,14)(5,16)(20,27)(21,29)(23,25)(24,26)(34,39)(35,42)(40,45)(43,48);(1,3,6,12)(2,8,7,11)(4,15,14,10)(5,13,16,9)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31)(33,34)(35,36)(37,40)(38,39)(41,42)(43,44)(45,46)(47,48);(1,4)(2,9)(3,10)(5,11)(6,14)(7,13)(8,16)(12,15)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,48)(46,47);(1,5,6,16)(2,10,7,15)(3,13,12,9)(4,11,14,8)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,37,38,46)(34,40,39,45)(35,43,42,48)(36,44,41,47)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9414080);