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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3844 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)(3,11)(4,13)(5,15)(6,7)(8,12)(9,14)(10,16)(19,25)(23,26)(27,31)(30,32)(35,37)(39,41)(43,45)(47,48);(2,7)(5,16)(8,11)(9,13)(17,18)(19,26)(20,28)(21,22)(23,25)(24,29)(27,30)(31,32)(33,34,38,42)(35,39,37,41)(36,44,46,40)(43,48,45,47);(2,7)(3,12)(9,13)(10,15)(18,22)(23,26)(24,28)(30,32)(33,35,38,37)(34,39,42,41)(36,43,46,45)(40,47,44,48);(1,3)(2,8)(4,10)(5,13)(6,12)(7,11)(9,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32);(1,4)(2,9)(3,10)(5,11)(6,14)(7,13)(8,16)(12,15)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32)(33,36,38,46)(34,40,42,44)(35,43,37,45)(39,47,41,48);(1,5,6,16)(2,10,7,15)(3,13,12,9)(4,11,14,8)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,37,38,35)(34,41,42,39)(36,45,46,43)(40,48,44,47)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8663334);