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 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3974 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)(3,5,9,14)(4,11,13,8)(10,16,15,12)(17,18,23,22)(19,26,24,20)(21,29,30,25)(27,28,31,32)(33,34,40,39)(35,45,47,41)(36,48,46,42)(37,44,38,43);(1,2,6,7)(3,5,9,14)(4,8,13,11)(10,12,15,16)(17,19,23,24)(18,20,22,26)(21,27,30,31)(25,28,29,32)(33,35)(34,41)(36,37)(38,46)(39,45)(40,47)(42,44)(43,48);(2,7)(5,14)(8,11)(12,16)(17,20,23,26)(18,24,22,19)(21,28,30,32)(25,31,29,27)(33,36,40,46)(34,42,39,48)(35,37,47,38)(41,44,45,43);(1,3,6,9)(2,5,7,14)(4,10,13,15)(8,12,11,16)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,32)(30,31)(33,37,40,38)(34,43,39,44)(35,36,47,46)(41,48,45,42);(1,4,6,13)(2,8,7,11)(3,10,9,15)(5,12,14,16)(17,21,23,30)(18,25,22,29)(19,27,24,31)(20,28,26,32)(33,38,40,37)(34,44,39,43)(35,46,47,36)(41,42,45,48);(1,5)(2,9)(3,7)(4,12)(6,14)(8,15)(10,11)(13,16)(17,22,23,18)(19,26,24,20)(21,29,30,25)(27,32,31,28)(33,39,40,34)(35,45,47,41)(36,42,46,48)(37,44,38,43)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=g,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8095418);