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 | #38593 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3911 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,22,27)(19,26,29,23)(20,25,21,24)(28,31,30,32)(33,34,39,36)(35,40,37,41)(38,42,46,44)(43,47,45,48);(1,2,6,7)(3,5,9,14)(4,11,13,8)(10,16,15,12)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,32)(26,27)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(2,7)(5,14)(8,11)(12,16)(17,20,22,21)(18,24,27,25)(19,28,29,30)(23,31,26,32)(33,35,39,37)(34,41,36,40)(38,43,46,45)(42,48,44,47);(1,3,6,9)(2,5,7,14)(4,10,13,15)(8,12,11,16)(17,21,22,20)(18,25,27,24)(19,28,29,30)(23,31,26,32)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,4,6,13)(2,8,7,11)(3,10,9,15)(5,12,14,16)(17,18)(19,26)(20,24)(21,25)(22,27)(23,29)(28,32)(30,31)(33,37,39,35)(34,40,36,41)(38,45,46,43)(42,47,44,48);(1,5)(2,9)(3,7)(4,12)(6,14)(8,15)(10,11)(13,16)(17,19)(18,26)(20,30)(21,28)(22,29)(23,27)(24,31)(25,32)(33,38,39,46)(34,42,36,44)(35,43,37,45)(40,47,41,48)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=h,d^2=hi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8137891);