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 | |
4395 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,16)(14,15)(18,19)(23,25)(24,26)(30,31)(33,34)(35,38)(36,40)(37,39)(41,42)(43,46)(44,45)(47,48);(2,7)(4,15)(8,11)(10,13)(17,18,22,19)(20,23,27,25)(21,24,29,26)(28,30,32,31)(33,35)(34,38)(36,43)(37,42)(39,41)(40,46)(44,48)(45,47);(2,7)(3,12)(4,15)(5,16)(20,27)(21,29)(23,25)(24,26)(34,39)(35,42)(36,44)(46,47);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31)(33,36,37,44)(34,40,39,45)(35,43,42,48)(38,46,41,47);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,34)(35,41)(36,45)(37,39)(38,42)(40,44)(43,46)(47,48);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,35)(34,41)(36,43)(37,42)(38,39)(40,47)(44,48)(45,46)Polycyclic presentation:
2^9|b^2=ghi,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9414032);