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 | |
4119 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,11,7)(4,12,13,8)(10,14,16,15)(17,18,23,25)(19,24,27,32)(20,22,30,31)(21,29,28,26)(34,39)(35,43)(37,41)(44,48);(2,6)(7,9)(8,12)(14,15)(17,19)(18,24)(20,21)(22,29)(23,27)(25,32)(26,31)(28,30)(33,34)(35,40)(36,41)(37,42)(38,39)(43,46)(44,47)(45,48);(2,6)(7,9)(8,12)(14,15)(18,25)(19,27)(20,30)(26,29)(35,43)(40,46)(44,48)(45,47);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20)(18,22)(19,28)(21,27)(23,30)(24,26)(25,31)(29,32)(33,35,38,43)(34,40,39,46)(36,44,42,48)(37,45,41,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,15)(13,16)(17,21,23,28)(18,26,25,29)(19,20,27,30)(22,24,31,32)(33,36,38,42)(34,41,39,37)(35,44,43,48)(40,47,46,45);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,22)(18,20)(19,29)(21,24)(23,31)(25,30)(26,27)(28,32)(33,37,38,41)(34,42,39,36)(35,45,43,47)(40,48,46,44)Polycyclic presentation:
2^9|a^2=h,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014436);