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 | |
4041 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)(3,9)(4,10)(5,13)(6,7)(8,14)(11,12)(15,16)(21,29)(26,30)(27,32)(28,31)(33,34,38,42)(35,41,37,39)(36,40,44,47)(43,48,45,46);(2,7)(3,10)(8,13)(11,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,35,38,37)(34,39,42,41)(36,43,44,45)(40,46,47,48);(2,7)(4,9)(8,13)(12,15)(18,23)(20,24)(26,30)(28,31)(36,44)(40,47)(43,45)(46,48);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,15)(14,16)(17,19,22,25)(18,24,23,20)(21,27,29,32)(26,31,30,28)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27)(33,36,38,44)(34,40,42,47)(35,43,37,45)(39,46,41,48);(1,5,6,14)(2,8,7,13)(3,11,10,16)(4,12,9,15)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)(33,37,38,35)(34,41,42,39)(36,45,44,43)(40,48,47,46)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=i,e^2=ghi,f^2=g,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, 8028632);