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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3967 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,5,9,7)(4,12,13,8)(11,14,16,15)(17,18)(19,22)(20,25)(21,30)(23,24)(26,28)(27,32)(29,31)(33,34,39,44)(35,45,47,40)(36,38,43,41)(37,48,46,42);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,19,23,27)(18,22,24,32)(20,21,29,28)(25,30,31,26)(33,35)(34,40)(36,37)(38,42)(39,47)(41,48)(43,46)(44,45);(4,13)(8,12)(11,16)(14,15)(18,24)(19,27)(20,29)(26,30)(33,36,39,43)(34,41,44,38)(35,37,47,46)(40,48,45,42);(1,4)(2,8)(3,11)(5,15)(6,13)(7,14)(9,16)(10,12)(17,20,23,29)(18,25,24,31)(19,28,27,21)(22,26,32,30)(33,36)(34,41)(35,46)(37,47)(38,44)(39,43)(40,42)(45,48);(1,3)(2,5)(4,11)(6,9)(7,10)(8,15)(12,14)(13,16)(17,21,23,28)(18,26,24,30)(19,29,27,20)(22,25,32,31)(33,37,39,46)(34,42,44,48)(35,36,47,43)(38,45,41,40);(1,5)(2,9)(3,10)(4,14)(6,7)(8,11)(12,16)(13,15)(17,22,23,32)(18,19,24,27)(20,30,29,26)(21,31,28,25)(33,38,39,41)(34,43,44,36)(35,42,47,48)(37,40,46,45)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=hi,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8627675);