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 | |
4043 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,9)(3,7,10,6)(4,12,13,8)(11,14,16,15)(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32)(33,34,40,44)(35,37,38,41)(36,45,48,42)(39,47,46,43);(3,10)(6,7)(11,16)(14,15)(18,24)(19,20)(26,29)(27,28)(33,35,40,38)(34,41,44,37)(36,43,48,47)(39,45,46,42);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32)(33,36)(34,42)(35,43)(37,46)(38,47)(39,41)(40,48)(44,45);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,37,40,41)(34,35,44,38)(36,46,48,39)(42,43,45,47);(1,2)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32)(33,38,40,35)(34,37,44,41)(36,47,48,43)(39,42,46,45);(1,4,5,13)(2,8,9,12)(3,11,10,16)(6,14,7,15)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)(33,39,40,46)(34,43,44,47)(35,45,38,42)(36,41,48,37)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=ghi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8046669);