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 |
| prime radical | #2410 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4091 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,4,11,7)(5,13,15,9)(10,16,14,12)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,28)(29,32)(33,34,39,42)(35,40,37,36)(38,46,47,41)(43,44,45,48);(1,3,6,11)(2,7,8,4)(5,14,15,10)(9,12,13,16)(17,19)(18,23)(20,21)(22,27)(24,30)(25,32)(26,28)(29,31)(33,34,39,42)(35,40,37,36)(38,46,47,41)(43,44,45,48);(2,8)(4,7)(9,13)(12,16)(19,27)(20,29)(23,28)(24,32)(33,35,39,37)(34,36,42,40)(38,43,47,45)(41,44,46,48);(1,4)(2,3)(5,12)(6,7)(8,11)(9,10)(13,14)(15,16)(17,20,22,29)(18,24,26,32)(19,21,27,31)(23,30,28,25)(33,36)(34,35)(37,42)(38,44)(39,40)(41,43)(45,46)(47,48);(1,3)(2,7)(4,8)(5,10)(6,11)(9,16)(12,13)(14,15)(17,21)(18,25)(19,20)(22,31)(23,32)(24,28)(26,30)(27,29)(33,37,39,35)(34,40,42,36)(38,45,47,43)(41,48,46,44);(1,5)(2,9)(3,10)(4,12)(6,15)(7,16)(8,13)(11,14)(17,18)(19,28)(20,24)(21,25)(22,26)(23,27)(29,32)(30,31)(33,38,39,47)(34,41,42,46)(35,43,37,45)(36,44,40,48)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=ghi,e^2=gh,f^2=gh,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, 8746443);