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 | |
3836 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)(3,8)(4,12)(5,10)(6,7)(9,14)(11,13)(15,16)(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32)(33,34,39,36)(35,42,38,40)(37,44,46,41)(43,47,45,48);(2,7)(3,11)(9,12)(10,16)(18,24)(19,20)(26,29)(27,28)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(2,7)(3,11)(9,12)(10,16)(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32)(33,36,39,34)(35,40,38,42)(37,44,46,41)(43,47,45,48);(1,3,6,11)(2,8,7,13)(4,10,14,16)(5,9,15,12)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,4,6,14)(2,9,7,12)(3,10,11,16)(5,8,15,13)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32)(33,35,39,38)(34,42,36,40)(37,43,46,45)(41,48,44,47);(1,5,6,15)(2,10,7,16)(3,12,11,9)(4,13,14,8)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)(33,38,39,35)(34,40,36,42)(37,45,46,43)(41,47,44,48)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=i,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8648907);