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 | |
4326 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)(3,9)(4,12)(5,10)(6,7)(8,13)(11,15)(14,16)(19,25)(23,26)(27,31)(30,32)(33,34,38,39)(35,40,43,37)(36,44,47,41)(42,46,45,48);(2,7)(4,13)(5,9)(11,14)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31)(34,39)(37,40)(41,44)(46,48);(2,7)(3,10)(8,12)(11,14)(18,22)(23,26)(24,28)(30,32)(33,35,38,43)(34,40,39,37)(36,42,47,45)(41,48,44,46);(1,3)(2,5)(4,14)(6,10)(7,9)(8,16)(11,13)(12,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35)(34,37)(36,45)(38,43)(39,40)(41,48)(42,47)(44,46);(1,4)(2,8)(3,11)(5,15)(6,13)(7,12)(9,16)(10,14)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32)(33,36)(34,41)(35,42)(37,46)(38,47)(39,44)(40,48)(43,45);(1,5)(2,3)(4,15)(6,9)(7,10)(8,11)(12,14)(13,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,37)(34,35)(36,46)(38,40)(39,43)(41,42)(44,45)(47,48)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8761490);