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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3854 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)(3,9)(4,7)(5,8)(6,12)(10,14)(11,15)(13,16)(17,18,22,24)(19,25,28,23)(20,29,30,21)(26,27,32,31)(34,37)(35,43)(36,46)(42,48);(2,4)(8,9)(10,12)(15,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,25)(29,31)(30,32)(33,34,39,37)(35,44,43,40)(36,47,46,41)(38,42,45,48);(1,3,7,5)(2,8,4,9)(6,11,14,13)(10,15,12,16)(17,18,22,24)(19,23,28,25)(20,29,30,21)(26,31,32,27)(33,35)(34,40)(36,45)(37,44)(38,46)(39,43)(41,48)(42,47);(1,4,7,2)(3,8,5,9)(6,12,14,10)(11,15,13,16)(17,20,22,30)(18,21,24,29)(19,26,28,32)(23,27,25,31)(33,36,39,46)(34,41,37,47)(35,45,43,38)(40,48,44,42);(1,5,7,3)(2,9,4,8)(6,13,14,11)(10,16,12,15)(17,21)(18,20)(19,27)(22,29)(23,26)(24,30)(25,32)(28,31)(33,37,39,34)(35,44,43,40)(36,47,46,41)(38,48,45,42);(1,6,7,14)(2,10,4,12)(3,11,5,13)(8,15,9,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,25)(29,31)(30,32)(33,38,39,45)(34,42,37,48)(35,46,43,36)(40,47,44,41)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=gi,d^2=g,e^2=h,f^2=h,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, 8066254);