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 | |
4079 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,11)(3,10,12,7)(4,13,14,8)(5,9,15,16)(17,18,23,20)(19,26,22,24)(21,25,29,28)(27,32,30,31)(33,34,39,37)(35,38,42,40)(36,44,46,41)(43,47,48,45);(1,3,6,12)(2,7,11,10)(4,9,14,16)(5,8,15,13)(21,29)(25,28)(27,30)(31,32)(34,37)(35,42)(41,44)(43,48);(4,14)(5,15)(8,13)(9,16)(17,19,23,22)(18,24,20,26)(21,27,29,30)(25,31,28,32)(33,35,39,42)(34,40,37,38)(36,43,46,48)(41,47,44,45);(1,4)(2,8)(3,9)(5,10)(6,14)(7,15)(11,13)(12,16)(17,20,23,18)(19,26,22,24)(21,28,29,25)(27,32,30,31)(33,36,39,46)(34,41,37,44)(35,43,42,48)(38,45,40,47);(1,5)(2,9)(3,8)(4,10)(6,15)(7,14)(11,16)(12,13)(17,21,23,29)(18,25,20,28)(19,27,22,30)(24,31,26,32)(33,37,39,34)(35,38,42,40)(36,44,46,41)(43,45,48,47);(1,3)(2,10)(4,9)(5,8)(6,12)(7,11)(13,15)(14,16)(17,22,23,19)(18,26,20,24)(21,30,29,27)(25,32,28,31)(33,38)(34,35)(36,45)(37,42)(39,40)(41,43)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=h,d^2=h,e^2=h,f^2=hi,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, 8718404);