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 | |
4065 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,7,10,4)(5,8,14,16)(11,13,15,12)(17,18,22,23)(19,25,28,24)(20,21,29,32)(26,27,30,31)(33,34,38,43)(35,39,45,47)(36,41,46,40)(37,44,48,42);(1,3)(2,7)(4,9)(5,11)(6,10)(8,13)(12,16)(14,15)(18,23)(20,29)(24,25)(26,30)(33,35,38,45)(34,39,43,47)(36,42,46,44)(37,40,48,41);(3,10)(4,7)(11,15)(12,13)(17,19,22,28)(18,24,23,25)(20,26,29,30)(21,31,32,27)(33,36,38,46)(34,40,43,41)(35,44,45,42)(37,47,48,39);(1,4,6,7)(2,3,9,10)(5,13,14,12)(8,15,16,11)(17,19)(18,24)(20,30)(21,27)(22,28)(23,25)(26,29)(31,32)(33,36)(34,41)(35,42)(37,39)(38,46)(40,43)(44,45)(47,48);(1,5)(2,8)(3,11)(4,12)(6,14)(7,13)(9,16)(10,15)(17,20)(18,21)(19,26)(22,29)(23,32)(24,31)(25,27)(28,30)(33,35)(34,39)(36,44)(37,40)(38,45)(41,48)(42,46)(43,47);(1,3)(2,7)(4,9)(5,11)(6,10)(8,13)(12,16)(14,15)(17,21)(18,20)(19,27)(22,32)(23,29)(24,30)(25,26)(28,31)(33,37,38,48)(34,42,43,44)(35,40,45,41)(36,39,46,47)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=h,d^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8819406);