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 | |
3748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,12,6,8)(4,10,15,14)(5,11,16,13)(17,18)(19,25)(20,26)(21,29)(22,23)(24,30)(27,28)(31,32)(35,38)(40,42)(43,45)(47,48);(1,3,7,6)(2,8,9,12)(4,13,15,11)(5,10,16,14)(18,23)(19,26)(24,29)(27,32)(33,34,39,36)(35,40,38,42)(37,44,46,41)(43,48,45,47);(2,9)(4,15)(8,12)(11,13)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,31)(30,32)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,4)(2,10)(3,13)(5,12)(6,11)(7,15)(8,16)(9,14)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31)(33,36,39,34)(35,42,38,40)(37,44,46,41)(43,48,45,47);(1,5,7,16)(2,11,9,13)(3,14,6,10)(4,8,15,12)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32)(33,37,39,46)(34,41,36,44)(35,43,38,45)(40,47,42,48);(1,6,7,3)(2,12,9,8)(4,11,15,13)(5,14,16,10)(17,21,22,30)(18,24,23,29)(19,27,26,32)(20,28,25,31)(33,38,39,35)(34,42,36,40)(37,45,46,43)(41,48,44,47)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=gh,d^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113833);