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 | |
4037 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)(4,12)(5,10)(6,7)(8,13)(11,14)(15,16)(20,28)(24,29)(26,32)(30,31)(33,34,39,40)(35,37,36,41)(38,46,47,42)(43,48,44,45);(2,7)(3,10)(4,13)(14,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35)(34,37)(36,39)(38,43)(40,41)(42,45)(44,47)(46,48);(2,7)(3,10)(8,12)(11,16)(18,22)(23,25)(24,29)(30,31)(34,40)(35,36)(42,46)(43,44);(1,3)(2,5)(4,11)(6,10)(7,9)(8,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36,39,35)(34,41,40,37)(38,44,47,43)(42,48,46,45);(1,4,6,13)(2,8,7,12)(3,11,10,16)(5,14,9,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,37,39,41)(34,35,40,36)(38,45,47,48)(42,43,46,44);(1,5)(2,3)(4,14)(6,9)(7,10)(8,11)(12,16)(13,15)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,47)(34,42,40,46)(35,43,36,44)(37,45,41,48)Polycyclic presentation:
2^9|a^2=gh,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8019549);