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 | |
3977 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)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32)(33,34)(35,39)(36,42)(37,41)(38,40)(43,45)(44,48)(46,47);(1,2,5,8)(3,6,11,9)(4,7,13,16)(10,14,12,15)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(34,35)(37,45)(40,42)(44,46);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,18,22,25)(19,23,20,26)(21,29,30,24)(27,32,28,31)(33,35,39,34)(36,42,38,40)(37,43,45,41)(44,48,46,47);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,13)(15,16)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,36,39,38)(34,40,35,42)(37,44,45,46)(41,47,43,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,32)(29,30)(33,37,39,45)(34,41,35,43)(36,44,38,46)(40,47,42,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,21,22,30)(18,24,25,29)(19,27,20,28)(23,31,26,32)(33,38,39,36)(34,42,35,40)(37,46,45,44)(41,48,43,47)Polycyclic presentation:
2^9|b^2=gi,c^2=g,d^2=h,e^2=i,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, 8961187);