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 | |
4058 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)(3,5)(4,12)(6,7)(8,14)(9,10)(11,16)(13,15)(17,18,23,21)(19,26,25,22)(20,28,29,24)(27,30,31,32)(35,36)(39,40)(43,44)(46,47);(1,3)(2,5)(4,11)(6,9)(7,10)(8,13)(12,16)(14,15)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,32)(29,31)(33,34,38,42)(35,39,36,40)(37,41,45,48)(43,46,44,47);(2,7)(3,9)(8,12)(11,15)(20,29)(24,28)(27,31)(30,32)(33,35,38,36)(34,39,42,40)(37,43,45,44)(41,46,48,47);(1,2)(3,10)(4,12)(5,9)(6,7)(8,14)(11,13)(15,16)(17,20,23,29)(18,24,21,28)(19,27,25,31)(22,30,26,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,4,6,14)(2,8,7,12)(3,11,9,15)(5,13,10,16)(17,21,23,18)(19,26,25,22)(20,24,29,28)(27,30,31,32)(33,36,38,35)(34,40,42,39)(37,44,45,43)(41,47,48,46);(1,5,6,10)(2,9,7,3)(4,13,14,16)(8,15,12,11)(17,22,23,26)(18,25,21,19)(20,30,29,32)(24,31,28,27)(33,37,38,45)(34,41,42,48)(35,43,36,44)(39,46,40,47)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=hi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066444);