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 | #53015 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,29)(28,31)(30,32)(34,37)(35,36)(42,45)(43,44);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19)(18,21)(20,27)(22,25)(23,30)(24,26)(28,32)(29,31)(33,34,39,37)(35,41,36,40)(38,42,46,45)(43,48,44,47);(3,10)(6,7)(11,16)(14,15)(20,28)(23,31)(27,32)(29,30)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20)(18,23)(19,27)(21,30)(22,28)(24,29)(25,32)(26,31)(33,36,39,35)(34,41,37,40)(38,44,46,43)(42,48,45,47);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,19)(18,24)(20,27)(21,26)(22,25)(23,29)(28,32)(30,31)(33,37,39,34)(35,41,36,40)(38,45,46,42)(43,48,44,47);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,21,22,24)(18,25,26,19)(20,29,28,30)(23,27,31,32)(33,38,39,46)(34,42,37,45)(35,43,36,44)(40,47,41,48)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8635740);