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 | |
4243 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)(3,9)(4,12)(5,11)(6,7)(8,13)(10,15)(14,16)(33,34,37,41)(35,38,44,47)(36,40,45,39)(42,48,43,46);(1,3,6,11)(2,5,7,9)(4,10,13,14)(8,15,12,16)(17,18,22,23)(19,20,24,28)(21,25,29,30)(26,27,31,32)(33,35,37,44)(34,38,41,47)(36,43,45,42)(39,48,40,46);(2,7)(4,13)(5,9)(10,14)(18,23)(19,24)(21,29)(27,32)(33,36,37,45)(34,39,41,40)(35,42,44,43)(38,46,47,48);(1,4)(2,8)(3,10)(5,15)(6,13)(7,12)(9,16)(11,14)(17,19)(18,20)(21,26)(22,24)(23,28)(25,27)(29,31)(30,32)(33,36)(34,40)(35,43)(37,45)(38,46)(39,41)(42,44)(47,48);(1,3)(2,9)(4,14)(5,7)(6,11)(8,15)(10,13)(12,16)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,30)(29,32)(33,35)(34,38)(36,42)(37,44)(39,46)(40,48)(41,47)(43,45);(1,5,6,9)(2,3,7,11)(4,15,13,16)(8,10,12,14)(17,21)(18,25)(19,26)(20,27)(22,29)(23,30)(24,31)(28,32)(33,34)(35,38)(36,40)(37,41)(39,45)(42,48)(43,46)(44,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8815607);