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 | |
4124 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)(3,4)(5,13)(6,7)(8,10)(9,14)(11,16)(12,15)(20,28)(23,30)(26,32)(29,31)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(1,2)(3,10)(4,8)(5,9)(6,7)(11,16)(12,15)(13,14)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(2,7)(3,8)(9,13)(11,15)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,16)(14,15)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,4)(2,8)(3,7)(5,12)(6,10)(9,15)(11,13)(14,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,5,6,14)(2,9,7,13)(3,11,8,15)(4,12,10,16)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=hi,f^2=h,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, 8064820);