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 | |
4107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,8,5,9)(6,12,14,10)(11,16,13,15)(18,23)(19,25)(24,30)(27,31)(33,34,38,39)(35,41,43,36)(37,45,46,40)(42,44,48,47);(1,3,7,5)(2,8,4,9)(6,11,14,13)(10,15,12,16)(17,18)(19,26)(20,25)(21,24)(22,23)(27,32)(28,31)(29,30)(33,35,38,43)(34,36,39,41)(37,42,46,48)(40,44,45,47);(2,4)(3,5)(6,14)(15,16)(19,25)(20,26)(21,29)(24,30)(34,39)(36,41)(37,46)(42,48);(1,4,7,2)(3,8,5,9)(6,12,14,10)(11,15,13,16)(17,19,22,25)(18,20,23,26)(21,27,29,31)(24,28,30,32)(33,34)(35,41)(36,43)(37,40)(38,39)(42,47)(44,48)(45,46);(1,5,7,3)(2,9,4,8)(6,13,14,11)(10,16,12,15)(17,20)(18,19)(21,28)(22,26)(23,25)(24,27)(29,32)(30,31)(33,36,38,41)(34,35,39,43)(37,44,46,47)(40,42,45,48);(1,6)(2,10)(3,11)(4,12)(5,13)(7,14)(8,15)(9,16)(17,21,22,29)(18,24,23,30)(19,27,25,31)(20,28,26,32)(33,37,38,46)(34,40,39,45)(35,42,43,48)(36,44,41,47)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8966444);