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 | |
4108 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,21)(22,28)(23,25)(26,27)(29,32)(30,31)(33,34,39,40)(35,41,36,38)(37,45,47,42)(43,46,44,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19,23,29)(18,24,25,32)(20,31,27,22)(21,30,26,28)(34,40)(35,36)(42,45)(43,44);(3,10)(6,7)(11,16)(14,15)(18,25)(19,29)(20,27)(28,30)(33,35,39,36)(34,38,40,41)(37,43,47,44)(42,46,45,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,23,27)(18,26,25,21)(19,22,29,31)(24,30,32,28)(33,36,39,35)(34,41,40,38)(37,44,47,43)(42,48,45,46);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21,23,26)(18,27,25,20)(19,30,29,28)(22,32,31,24)(33,37,39,47)(34,42,40,45)(35,43,36,44)(38,46,41,48);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,22)(18,28)(19,20)(21,24)(23,31)(25,30)(26,32)(27,29)(33,38,39,41)(34,35,40,36)(37,46,47,48)(42,43,45,44)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,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, 8635530);