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 | #55497 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3983 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,13)(10,16)(14,15)(17,18)(19,26)(20,23)(21,28)(22,24)(25,30)(27,31)(29,32)(33,34,39,38)(35,42,37,40)(36,41,44,46)(43,48,45,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(36,44)(41,46)(43,45)(47,48);(2,6)(7,9)(8,13)(14,15)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,35,39,37)(34,40,38,42)(36,43,44,45)(41,47,46,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,23,18)(19,24,22,26)(21,28,30,25)(27,31,29,32)(33,36,39,44)(34,41,38,46)(35,43,37,45)(40,47,42,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,37,39,35)(34,42,38,40)(36,45,44,43)(41,48,46,47);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,22,23,19)(18,24,20,26)(21,29,30,27)(25,31,28,32)(33,38,39,34)(35,42,37,40)(36,46,44,41)(43,48,45,47)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,d^2=h,e^2=gh,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^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8019626);