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 | |
3850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,11)(8,13)(9,15)(14,16)(17,18,23,26)(19,25,20,24)(21,22,31,32)(27,30,29,28)(33,34,40,43)(35,44,39,41)(36,38,37,42)(45,47,46,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,19,23,20)(18,24,26,25)(21,27,31,29)(22,28,32,30)(33,35,40,39)(34,41,43,44)(36,46,37,45)(38,47,42,48);(2,6)(7,12)(8,13)(14,16)(17,19,23,20)(18,25,26,24)(21,29,31,27)(22,28,32,30)(33,36,40,37)(34,42,43,38)(35,45,39,46)(41,47,44,48);(1,3,5,10)(2,7,6,12)(4,9,11,15)(8,14,13,16)(17,20,23,19)(18,25,26,24)(21,29,31,27)(22,30,32,28)(33,37,40,36)(34,38,43,42)(35,46,39,45)(41,48,44,47);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,21)(18,22)(19,27)(20,29)(23,31)(24,28)(25,30)(26,32)(33,38)(34,36)(35,47)(37,43)(39,48)(40,42)(41,46)(44,45);(1,4)(2,8)(3,9)(5,11)(6,13)(7,14)(10,15)(12,16)(17,22)(18,21)(19,28)(20,30)(23,32)(24,27)(25,29)(26,31)(33,39,40,35)(34,41,43,44)(36,46,37,45)(38,48,42,47)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,d^2=ghi,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^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8025770);