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 | |
4056 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)(19,28)(20,25)(21,31)(22,29)(23,27)(24,30)(26,32)(33,34,38,41)(35,42,40,37)(36,44,46,39)(43,45,47,48);(3,10)(6,7)(11,16)(14,15)(17,19)(18,24)(20,22)(21,29)(23,30)(25,26)(27,28)(31,32)(33,35,38,40)(34,37,41,42)(36,43,46,47)(39,45,44,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,20,23,31)(18,25,27,21)(19,22,30,32)(24,26,28,29)(33,36)(34,39)(35,43)(37,45)(38,46)(40,47)(41,44)(42,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,21)(18,20)(19,29)(22,24)(23,25)(26,30)(27,31)(28,32)(33,35)(34,37)(36,43)(38,40)(39,45)(41,42)(44,48)(46,47);(1,2)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,25)(21,31)(22,26)(23,27)(28,30)(29,32)(33,37)(34,40)(35,41)(36,45)(38,42)(39,47)(43,44)(46,48);(1,4,5,13)(2,8,9,12)(3,11,10,16)(6,14,7,15)(17,22,23,32)(18,26,27,29)(19,20,30,31)(21,24,25,28)(33,36)(34,39)(35,43)(37,45)(38,46)(40,47)(41,44)(42,48)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=ghi,f^2=gh,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, 8046677);