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 | |
4138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,11)(8,13)(9,15)(14,16)(17,18,22,23)(19,21,26,24)(20,28,30,25)(27,31,32,29)(33,34,39,37)(35,40,46,44)(36,42,38,41)(43,48,45,47);(2,6)(7,12)(8,13)(14,16)(18,23)(19,26)(25,28)(27,32)(34,37)(40,44)(41,42)(47,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,35,39,46)(34,40,37,44)(36,43,38,45)(41,47,42,48);(1,3,5,10)(2,7,6,12)(4,9,11,15)(8,14,13,16)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29)(33,36,39,38)(34,41,37,42)(35,43,46,45)(40,47,44,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,37,39,34)(35,44,46,40)(36,42,38,41)(43,48,45,47);(1,4)(2,8)(3,9)(5,11)(6,13)(7,14)(10,15)(12,16)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)(33,38,39,36)(34,42,37,41)(35,45,46,43)(40,48,44,47)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8016454);