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 | #56651 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3978 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,15)(13,16)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30);(2,7)(3,11)(4,14)(13,15)(17,19,22,27)(18,23,25,20)(21,26,29,32)(24,30,31,28)(33,34,38,39)(35,37,41,46)(36,43,45,40)(42,48,47,44);(2,7)(5,9)(8,10)(13,15)(21,29)(24,31)(26,32)(28,30)(34,39)(37,46)(40,43)(44,48);(1,3,6,11)(2,8,7,10)(4,12,14,16)(5,13,9,15)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,30)(31,32)(33,35,38,41)(34,37,39,46)(36,42,45,47)(40,44,43,48);(1,4)(2,9)(3,12)(5,7)(6,14)(8,15)(10,13)(11,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,36,38,45)(34,40,39,43)(35,42,41,47)(37,44,46,48);(1,5,6,9)(2,4,7,14)(3,13,11,15)(8,12,10,16)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)(33,37)(34,41)(35,39)(36,44)(38,46)(40,47)(42,43)(45,48)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,d^2=hi,e^2=i,f^2=g,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, 8134548);