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 | |
4132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,4)(3,9)(8,12)(10,15)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,28)(29,32)(33,34,38,39)(35,42,40,36)(37,45,46,41)(43,44,47,48);(1,2,6,4)(3,7,9,11)(5,12,13,8)(10,16,15,14)(17,19)(18,23)(20,21)(22,27)(24,30)(25,32)(26,28)(29,31)(34,39)(36,42)(37,46)(43,47);(3,9)(7,11)(10,15)(14,16)(19,27)(20,29)(23,28)(24,32)(33,35,38,40)(34,36,39,42)(37,43,46,47)(41,44,45,48);(1,3,6,9)(2,7,4,11)(5,10,13,15)(8,14,12,16)(17,20,22,29)(18,24,26,32)(19,21,27,31)(23,30,28,25)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,48)(46,47);(1,4,6,2)(3,7,9,11)(5,12,13,8)(10,14,15,16)(17,21)(18,25)(19,20)(22,31)(23,32)(24,28)(26,30)(27,29)(33,36)(34,40)(35,39)(37,44)(38,42)(41,47)(43,45)(46,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,18)(19,28)(20,24)(21,25)(22,26)(23,27)(29,32)(30,31)(33,37,38,46)(34,41,39,45)(35,43,40,47)(36,44,42,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,d^2=ghi,e^2=gh,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, 8746425);