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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4102 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18)(19,26)(20,23)(21,29)(22,24)(25,30)(27,31)(28,32)(34,38)(36,43)(39,45)(41,44);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,22,20)(18,23,24,26)(21,27,30,28)(25,31,29,32)(34,38)(39,45)(40,46)(47,48);(2,3)(7,9)(8,10)(14,15)(18,24)(23,26)(25,29)(31,32)(33,34,37,38)(35,39,42,45)(36,40,43,46)(41,47,44,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,18)(19,26)(20,23)(21,25)(22,24)(27,32)(28,31)(29,30)(33,34)(35,39)(36,40)(37,38)(41,47)(42,45)(43,46)(44,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20,22,19)(18,23,24,26)(21,27,30,28)(25,32,29,31)(33,35,37,42)(34,39,38,45)(36,44,43,41)(40,48,46,47);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21,22,30)(18,25,24,29)(19,27,20,28)(23,31,26,32)(33,36)(34,40)(35,41)(37,43)(38,46)(39,47)(42,44)(45,48)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9852729);