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 | |
3745 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18)(19,28)(20,24)(21,25)(22,26)(23,27)(29,30)(31,32)(33,34,39,37)(35,42,38,40)(36,41,46,45)(43,48,44,47);(1,3,6,11)(2,7,8,16)(4,14,10,12)(5,13,9,15)(17,19)(18,22)(20,29)(21,31)(23,26)(24,32)(25,30)(27,28)(33,35,39,38)(34,40,37,42)(36,44,46,43)(41,48,45,47);(2,8)(3,11)(4,10)(13,15)(17,20,23,21)(18,24,27,25)(19,29,26,31)(22,32,28,30)(33,35,39,38)(34,40,37,42)(36,44,46,43)(41,48,45,47);(1,4,6,10)(2,9,8,5)(3,12,11,14)(7,13,16,15)(17,21,23,20)(18,25,27,24)(19,29,26,31)(22,32,28,30)(33,36,39,46)(34,41,37,45)(35,43,38,44)(40,47,42,48);(1,3)(2,7)(4,12)(5,15)(6,11)(8,16)(9,13)(10,14)(17,22,23,28)(18,26,27,19)(20,30,21,32)(24,29,25,31)(33,37,39,34)(35,40,38,42)(36,45,46,41)(43,47,44,48);(1,5,6,9)(2,10,8,4)(3,13,11,15)(7,12,16,14)(17,18)(19,22)(20,24)(21,25)(23,27)(26,28)(29,32)(30,31)(33,38,39,35)(34,42,37,40)(36,44,46,43)(41,48,45,47)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=g,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9947660);