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 | |
3987 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,18)(19,26)(20,24)(21,25)(22,23)(27,30)(28,32)(29,31)(34,39)(35,37)(41,44)(43,45);(2,7)(4,8)(10,14)(12,15)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,32)(30,31)(33,34,38,39)(35,40,37,42)(36,44,46,41)(43,48,45,47);(3,9)(4,8)(5,13)(10,14)(18,22)(20,26)(21,29)(27,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,14)(13,16)(17,20,23,26)(18,24,22,19)(21,30,29,28)(25,27,31,32)(33,35,38,37)(34,40,39,42)(36,43,46,45)(41,47,44,48);(1,4,6,8)(2,9,7,3)(5,12,13,15)(10,16,14,11)(17,21,23,29)(18,25,22,31)(19,27,24,32)(20,28,26,30)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,5,6,13)(2,10,7,14)(3,11,9,16)(4,12,8,15)(17,22,23,18)(19,26,24,20)(21,31,29,25)(27,30,32,28)(33,37,38,35)(34,42,39,40)(36,45,46,43)(41,48,44,47)Polycyclic presentation:
2^9|b^2=i,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8961197);