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 | |
4063 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
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)(21,29)(24,26)(27,32)(30,31)(34,39)(37,45)(40,41)(43,44);(2,7)(3,9)(10,13)(11,16)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(2,7)(3,9)(10,13)(11,16)(18,19)(23,25)(24,26)(30,31)(33,35,38,36)(34,40,39,41)(37,43,45,44)(42,47,46,48);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,13)(14,16)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31)(33,34)(35,41)(36,40)(37,42)(38,39)(43,48)(44,47)(45,46);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,36,38,35)(34,41,39,40)(37,44,45,43)(42,48,46,47);(1,5,6,14)(2,10,7,13)(3,11,9,16)(4,12,8,15)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,45)(34,42,39,46)(35,43,36,44)(40,47,41,48)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^2=i,e^2=gi,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=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8074629);