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 | |
4120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,12,8)(4,13,15,9)(5,10,16,14)(17,18)(19,24)(20,22)(21,25)(23,27)(26,31)(28,29)(30,32)(33,34)(35,39)(36,44)(37,38)(40,46)(41,43)(42,48)(45,47);(2,7)(3,12)(9,13)(10,14)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,46)(45,47);(2,7)(5,16)(8,11)(9,13)(21,29)(25,28)(26,32)(30,31)(34,38)(39,41)(40,44)(47,48);(1,3)(2,8)(4,14)(5,9)(6,12)(7,11)(10,15)(13,16)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30)(33,35,37,43)(34,39,38,41)(36,45,46,42)(40,48,44,47);(1,4,6,15)(2,9,7,13)(3,10,12,14)(5,8,16,11)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,36,37,46)(34,40,38,44)(35,42,43,45)(39,47,41,48);(1,5,6,16)(2,10,7,14)(3,9,12,13)(4,8,15,11)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=hi,d^2=gh,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8231752);