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 |
| prime radical | #45408 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4081 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,15)(11,16)(13,14)(18,20)(23,26)(24,28)(30,32)(33,34,39,42)(35,40,37,36)(38,41,47,46)(43,48,45,44);(1,2,6,10)(3,5,9,7)(4,8,12,15)(11,13,16,14)(19,25)(23,26)(27,31)(30,32)(33,35,39,37)(34,36,42,40)(38,43,47,45)(41,44,46,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,36,39,40)(34,35,42,37)(38,44,47,48)(41,43,46,45);(1,3)(2,5)(4,11)(6,9)(7,10)(8,13)(12,16)(14,15)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32)(33,34)(35,36)(37,40)(38,46)(39,42)(41,47)(43,48)(44,45);(1,4,6,12)(2,8,10,15)(3,11,9,16)(5,13,7,14)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,37,39,35)(34,40,42,36)(38,45,47,43)(41,48,46,44);(1,5)(2,9)(3,10)(4,13)(6,7)(8,16)(11,15)(12,14)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,38,39,47)(34,41,42,46)(35,43,37,45)(36,44,40,48)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gh,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8352556);