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 | |
4260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,11,12,7)(4,13,14,9)(5,10,15,16)(17,18,23,25)(19,24,21,20)(22,26,31,32)(27,30,29,28)(33,34)(35,45)(36,42)(37,39)(38,43)(40,44)(41,48)(46,47);(1,3,6,12)(2,7,8,11)(4,10,14,16)(5,9,15,13)(17,19,23,21)(18,24,25,20)(22,27,31,29)(26,30,32,28)(33,35)(34,41)(36,46)(37,43)(38,42)(39,47)(40,48)(44,45);(2,8)(4,14)(7,11)(10,16)(18,25)(19,21)(26,32)(27,29)(33,36,40,37)(34,42,44,39)(35,46,48,43)(38,45,47,41);(1,3)(2,7)(4,10)(5,13)(6,12)(8,11)(9,15)(14,16)(17,20,23,24)(18,19,25,21)(22,30,31,28)(26,29,32,27)(33,37,40,36)(34,42,44,39)(35,43,48,46)(38,45,47,41);(1,4,6,14)(2,9,8,13)(3,10,12,16)(5,7,15,11)(17,21,23,19)(18,24,25,20)(22,29,31,27)(26,30,32,28)(33,38,40,47)(34,43,44,46)(35,42,48,39)(36,41,37,45);(1,5,6,15)(2,10,8,16)(3,9,12,13)(4,7,14,11)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)(33,39,40,42)(34,37,44,36)(35,47,48,38)(41,43,45,46)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=ghi,d^2=gh,e^2=gi,f^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, 8441873);