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 | #37587 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4049 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)(3,4)(5,13)(6,7)(8,9)(10,14)(11,15)(12,16)(17,18,22,24)(19,23,25,20)(21,29,30,26)(27,28,32,31)(33,34,38,42)(35,43,45,39)(36,40,46,44)(37,47,48,41);(2,7)(3,9)(10,13)(11,16)(17,19,22,25)(18,23,24,20)(21,27,30,32)(26,31,29,28)(33,35)(34,39)(36,41)(37,44)(38,45)(40,48)(42,43)(46,47);(2,7)(3,9)(10,13)(11,16)(18,24)(19,25)(26,29)(27,32)(33,34,38,42)(35,39,45,43)(36,40,46,44)(37,41,48,47);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,13)(14,16)(17,19)(18,23)(20,24)(21,27)(22,25)(26,31)(28,29)(30,32)(33,34)(35,39)(36,44)(37,41)(38,42)(40,46)(43,45)(47,48);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,20,22,23)(18,25,24,19)(21,28,30,31)(26,32,29,27)(33,36)(34,40)(35,41)(37,43)(38,46)(39,48)(42,44)(45,47);(1,5)(2,10)(3,11)(4,12)(6,14)(7,13)(8,15)(9,16)(17,21,22,30)(18,26,24,29)(19,27,25,32)(20,28,23,31)(33,37,38,48)(34,41,42,47)(35,44,45,40)(36,43,46,39)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066382);