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 | |
4323 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,8)(4,10)(5,7)(6,9)(11,15)(12,16)(13,14)(17,18,22,25)(19,23,28,21)(20,24,30,32)(26,31,27,29)(33,34)(35,43)(36,47)(37,48)(38,39)(40,46)(41,45)(42,44);(2,5)(8,10)(9,13)(15,16)(17,19,22,28)(18,23,25,21)(20,27,30,26)(24,29,32,31)(33,34,39,38)(35,40,46,43)(36,41,45,47)(37,48,44,42);(1,3,7,4)(2,8,5,10)(6,11,14,12)(9,15,13,16)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28)(33,35)(34,40)(36,44)(37,45)(38,43)(39,46)(41,42)(47,48);(1,4,7,3)(2,8,5,10)(6,11,14,12)(9,16,13,15)(17,20)(18,24)(19,27)(21,31)(22,30)(23,29)(25,32)(26,28)(33,36,39,45)(34,41,38,47)(35,44,46,37)(40,42,43,48);(1,5,7,2)(3,10,4,8)(6,13,14,9)(11,16,12,15)(17,19)(18,21)(20,26)(22,28)(23,25)(24,29)(27,30)(31,32)(33,37,39,44)(34,42,38,48)(35,45,46,36)(40,41,43,47);(1,6,7,14)(2,9,5,13)(3,11,4,12)(8,15,10,16)(17,21)(18,19)(20,29)(22,23)(24,26)(25,28)(27,32)(30,31)(33,38,39,34)(35,43,46,40)(36,41,45,47)(37,48,44,42)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=gh,d^2=g,e^2=g,f^2=g,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, 10326889);