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 | |
4073 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,9)(3,7,10,6)(4,12,13,8)(11,14,16,15)(17,18,23,22)(19,24,30,29)(20,26,21,25)(27,32,28,31)(33,34,38,41)(35,42,39,36)(37,45,46,40)(43,44,47,48);(3,10)(6,7)(11,16)(14,15)(17,19,23,30)(18,24,22,29)(20,27,21,28)(25,31,26,32)(33,35,38,39)(34,36,41,42)(37,43,46,47)(40,44,45,48);(1,3,5,10)(2,6,9,7)(4,11,13,16)(8,14,12,15)(17,20,23,21)(18,25,22,26)(19,27,30,28)(24,31,29,32)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,46)(47,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,36,38,42)(34,39,41,35)(37,44,46,48)(40,47,45,43);(1,2)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,21,23,20)(18,26,22,25)(19,28,30,27)(24,32,29,31)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,46)(47,48);(1,4,5,13)(2,8,9,12)(3,11,10,16)(6,14,7,15)(17,22,23,18)(19,29,30,24)(20,26,21,25)(27,32,28,31)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=gh,d^2=g,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8072397);