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 | |
4240 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)(17,18)(19,25)(20,28)(21,22)(23,26)(24,30)(27,31)(29,32)(35,41)(39,42)(43,47)(46,48);(2,7)(4,10)(5,14)(11,15)(18,22)(23,25)(24,28)(31,32)(33,34)(35,39)(36,44)(37,38)(40,45)(41,42)(43,48)(46,47);(2,7)(3,8)(9,13)(11,15)(19,26)(23,25)(27,29)(31,32)(34,38)(39,42)(40,44)(46,48);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,16)(14,15)(17,19,21,26)(18,23,22,25)(20,29,30,27)(24,32,28,31)(33,34)(35,42)(36,40)(37,38)(39,41)(43,48)(44,45)(46,47);(1,4)(2,8)(3,7)(5,12)(6,10)(9,15)(11,13)(14,16)(17,20,21,30)(18,24,22,28)(19,27,26,29)(23,31,25,32)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,48)(45,47);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,30)(29,32)(33,36,37,45)(34,40,38,44)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^9|a^2=h,d^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, 8783589);