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 | |
4131 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)(3,6)(4,11)(5,7)(8,13)(9,15)(10,16)(12,14)(18,23)(19,24)(25,29)(26,31)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,46)(45,47);(1,3,5,9)(2,6,7,15)(4,12,13,10)(8,16,11,14)(17,18,22,23)(19,20,24,27)(21,29,30,25)(26,32,31,28)(34,37)(36,44)(39,41)(42,45);(2,7)(3,9)(8,11)(10,12)(19,24)(20,27)(26,31)(28,32)(33,35,38,43)(34,39,37,41)(36,45,44,42)(40,48,46,47);(1,2)(3,6)(4,8)(5,7)(9,15)(10,14)(11,13)(12,16)(17,19,22,24)(18,20,23,27)(21,26,30,31)(25,28,29,32)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,45)(46,48);(1,3)(2,6)(4,10)(5,9)(7,15)(8,14)(11,16)(12,13)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,29)(30,32)(33,36)(34,40)(35,42)(37,46)(38,44)(39,47)(41,48)(43,45);(1,4,5,13)(2,8,7,11)(3,10,9,12)(6,14,15,16)(17,21,22,30)(18,25,23,29)(19,26,24,31)(20,28,27,32)(33,37,38,34)(35,41,43,39)(36,46,44,40)(42,48,45,47)Polycyclic presentation:
2^9|b^2=gh,c^2=hi,d^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8725207);