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 | |
3873 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)(3,5)(4,13)(6,14)(7,9)(8,12)(10,11)(15,16)(17,18,21,24)(19,25,27,22)(20,23,29,32)(26,31,28,30)(33,34,39,35)(36,43,46,40)(37,41,38,42)(44,48,45,47);(1,3,7,12)(2,8,9,5)(4,10,14,16)(6,11,13,15)(17,19)(18,22)(20,28)(21,27)(23,31)(24,25)(26,29)(30,32)(34,35)(40,43)(41,42)(47,48);(2,9)(4,14)(5,8)(10,16)(17,19,21,27)(18,22,24,25)(20,26,29,28)(23,30,32,31)(33,35,39,34)(36,43,46,40)(37,42,38,41)(44,48,45,47);(1,4,7,14)(2,6,9,13)(3,10,12,16)(5,15,8,11)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,29)(31,32)(33,36,39,46)(34,40,35,43)(37,44,38,45)(41,47,42,48);(1,5,7,8)(2,3,9,12)(4,11,14,15)(6,16,13,10)(17,20)(18,23)(19,26)(21,29)(22,30)(24,32)(25,31)(27,28)(33,37,39,38)(34,41,35,42)(36,44,46,45)(40,47,43,48);(1,6)(2,4)(3,11)(5,16)(7,13)(8,10)(9,14)(12,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32)(33,38,39,37)(34,42,35,41)(36,45,46,44)(40,48,43,47)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=i,d^2=hi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9575475);