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 | |
3873 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,12,8)(4,13,14,9)(5,10,15,16)(17,18)(19,24)(20,29)(21,26)(22,23)(25,30)(27,28)(31,32)(33,34,39,38)(35,42,40,36)(37,45,46,41)(43,44,47,48);(2,7)(3,12)(9,13)(10,16)(17,18)(19,27)(20,25)(21,26)(22,23)(24,28)(29,30)(31,32)(33,35,39,40)(34,36,38,42)(37,43,46,47)(41,44,45,48);(2,7)(3,12)(9,13)(10,16)(18,23)(19,28)(25,29)(26,32)(33,36,39,42)(34,40,38,35)(37,44,46,48)(41,47,45,43);(1,3,6,12)(2,8,7,11)(4,10,14,16)(5,9,15,13)(17,19)(18,24)(20,26)(21,29)(22,28)(23,27)(25,31)(30,32)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,4,6,14)(2,9,7,13)(3,10,12,16)(5,8,15,11)(17,20)(18,25)(19,26)(21,27)(22,30)(23,29)(24,31)(28,32)(33,37,39,46)(34,41,38,45)(35,43,40,47)(36,44,42,48);(1,5,6,15)(2,10,7,16)(3,13,12,9)(4,11,14,8)(17,21,22,31)(18,26,23,32)(19,25,28,29)(20,24,30,27)(33,38,39,34)(35,42,40,36)(37,45,46,41)(43,48,47,44)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=hi,e^2=g,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, 8627590);