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 | |
3953 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,9)(3,7,11,6)(4,12,13,8)(10,14,16,15)(17,18,22,25)(19,26,23,20)(21,24,30,32)(27,31,29,28)(35,38)(40,42)(43,45)(47,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,30)(31,32)(33,34,39,36)(35,40,38,42)(37,41,46,44)(43,47,45,48);(1,2)(3,7)(4,8)(5,9)(6,11)(10,15)(12,13)(14,16)(17,18,22,25)(19,20,23,26)(21,24,30,32)(27,28,29,31)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,2)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,19)(18,20)(21,29)(22,23)(24,31)(25,26)(27,30)(28,32)(33,36,39,34)(35,42,38,40)(37,41,46,44)(43,47,45,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,14)(13,16)(17,20)(18,23)(19,25)(21,28)(22,26)(24,29)(27,32)(30,31)(33,37,39,46)(34,41,36,44)(35,43,38,45)(40,47,42,48);(1,4,5,13)(2,8,9,12)(3,10,11,16)(6,14,7,15)(17,21,22,30)(18,24,25,32)(19,27,23,29)(20,28,26,31)(33,38,39,35)(34,42,36,40)(37,45,46,43)(41,48,44,47)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=gh,d^2=hi,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10358857);