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 | |
3922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(7,14)(10,16)(13,15)(17,18)(19,25)(20,23)(21,24)(22,28)(26,30)(27,31)(29,32)(34,38)(36,44)(40,42)(43,45);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(18,20)(24,25)(26,28)(31,32)(33,34,39,38)(35,40,37,42)(36,41,44,46)(43,47,45,48);(1,2,5,8)(3,6,11,9)(4,7,12,14)(10,13,16,15)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(33,35,39,37)(34,40,38,42)(36,45,44,43)(41,48,46,47);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,20,23,18)(19,24,21,25)(22,28,30,26)(27,31,29,32)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,21,23,19)(18,25,20,24)(22,29,30,27)(26,32,28,31)(33,37,39,35)(34,42,38,40)(36,43,44,45)(41,47,46,48);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,22,23,30)(18,26,20,28)(19,27,21,29)(24,31,25,32)(33,38,39,34)(35,42,37,40)(36,46,44,41)(43,48,45,47)Polycyclic presentation:
2^9|b^2=gh,c^2=gi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8064153);