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 | |
4340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,6,9,13)(4,7,12,11)(10,14,15,16)(17,18,23,25)(19,24,21,20)(22,26,31,32)(27,30,29,28)(34,40)(35,36)(37,45)(46,48);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,19,23,21)(18,24,25,20)(22,27,31,29)(26,30,32,28)(33,34,39,40)(35,38,36,43)(37,41,45,47)(42,46,44,48);(3,9)(6,13)(10,15)(14,16)(18,25)(19,21)(26,32)(27,29)(33,35,39,36)(34,38,40,43)(37,44,45,42)(41,48,47,46);(1,2)(3,6)(4,11)(5,8)(7,12)(9,13)(10,16)(14,15)(17,20,23,24)(18,19,25,21)(22,30,31,28)(26,29,32,27)(33,36,39,35)(34,38,40,43)(37,44,45,42)(41,46,47,48);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,21,23,19)(18,24,25,20)(22,29,31,27)(26,30,32,28)(33,37)(34,41)(35,42)(36,44)(38,46)(39,45)(40,47)(43,48);(1,4)(2,7)(3,10)(5,12)(6,14)(8,11)(9,15)(13,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)(33,38)(34,36)(35,40)(37,46)(39,43)(41,44)(42,47)(45,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=hi,d^2=h,e^2=i,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, 8441814);