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 | |
4201 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)(3,9)(4,10)(5,13)(6,7)(8,14)(11,12)(15,16)(17,18,22,23)(19,25,26,24)(20,29,31,21)(27,32,30,28)(33,34)(35,39)(36,40)(37,46)(38,42)(41,47)(43,44)(45,48);(2,7)(4,9)(5,14)(11,16)(18,23)(21,29)(24,25)(28,32)(33,35)(34,39)(36,37)(38,43)(40,46)(41,48)(42,44)(45,47);(2,7)(3,10)(8,13)(11,16)(19,26)(24,25)(27,30)(28,32)(35,43)(36,45)(39,44)(40,48);(1,3,6,10)(2,4,7,9)(5,11,14,16)(8,12,13,15)(17,19)(18,24)(20,30)(21,28)(22,26)(23,25)(27,31)(29,32)(33,36,38,45)(34,40,42,48)(35,37,43,47)(39,46,44,41);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,20)(18,21)(19,27)(22,31)(23,29)(24,32)(25,28)(26,30)(33,37)(34,41)(35,36)(38,47)(39,48)(40,44)(42,46)(43,45);(1,5,6,14)(2,8,7,13)(3,11,10,16)(4,12,9,15)(17,21)(18,20)(19,28)(22,29)(23,31)(24,30)(25,27)(26,32)(33,34)(35,44)(36,40)(37,41)(38,42)(39,43)(45,48)(46,47)Polycyclic presentation:
2^9|a^2=gh,d^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8745622);