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:
(1,2,6,7)(3,10,8,4)(5,13,14,9)(11,12,15,16)(19,25)(23,26)(27,31)(30,32)(33,34,38,42)(35,36,40,39)(37,45,47,41)(43,46,48,44);(2,7)(4,10)(5,14)(11,15)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31)(33,35,38,40)(34,39,42,36)(37,43,47,48)(41,46,45,44);(2,7)(3,8)(9,13)(11,15)(18,22)(23,26)(24,28)(30,32)(33,36,38,39)(34,40,42,35)(37,44,47,46)(41,48,45,43);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,16)(14,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,36)(34,35)(37,46)(38,39)(40,42)(41,48)(43,45)(44,47);(1,4)(2,8)(3,7)(5,12)(6,10)(9,15)(11,13)(14,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31)(33,37,38,47)(34,41,42,45)(35,43,40,48)(36,44,39,46);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,20,21,29)(18,24,22,28)(19,27,25,31)(23,30,26,32)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gh,e^2=gh,f^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, 8783396);