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 | |
4266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(7,9)(8,13)(14,15)(17,18)(19,25)(20,24)(21,26)(22,23)(27,32)(28,31)(29,30)(33,34,38,41)(35,39,43,48)(36,44,37,40)(42,47,45,46);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19,23,20)(18,24,22,25)(21,28,30,27)(26,32,29,31)(33,35,38,43)(34,39,41,48)(36,45,37,42)(40,47,44,46);(4,12)(8,13)(10,16)(14,15)(18,22)(24,25)(26,29)(31,32)(33,36,38,37)(34,40,41,44)(35,42,43,45)(39,46,48,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,20,23,19)(18,25,22,24)(21,27,30,28)(26,31,29,32)(33,37,38,36)(34,40,41,44)(35,42,43,45)(39,47,48,46);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,21,23,30)(18,26,22,29)(19,27,20,28)(24,31,25,32)(33,35)(34,39)(36,42)(37,45)(38,43)(40,46)(41,48)(44,47);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,22,23,18)(19,25,20,24)(21,29,30,26)(27,32,28,31)(33,34)(35,39)(36,44)(37,40)(38,41)(42,47)(43,48)(45,46)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=i,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8805743);