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 | |
4253 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)(17,18,23,26)(19,24,30,32)(20,25,28,29)(21,31,27,22)(33,34,40,44)(35,43,39,36)(37,46,48,41)(38,45,47,42);(2,5)(8,10)(9,13)(15,16)(17,19)(18,24)(20,21)(22,29)(23,30)(25,31)(26,32)(27,28)(33,35,40,39)(34,36,44,43)(37,45,48,42)(38,41,47,46);(1,3,7,4)(2,8,5,10)(6,11,14,12)(9,15,13,16)(17,20)(18,25)(19,27)(21,30)(22,24)(23,28)(26,29)(31,32)(33,36)(34,39)(35,44)(37,47)(38,48)(40,43)(41,42)(45,46);(1,4,7,3)(2,8,5,10)(6,11,14,12)(9,16,13,15)(17,20)(18,25)(19,21)(22,32)(23,28)(24,31)(26,29)(27,30)(33,37,40,48)(34,41,44,46)(35,45,39,42)(36,47,43,38);(1,5,7,2)(3,10,4,8)(6,13,14,9)(11,16,12,15)(17,21)(18,22)(19,28)(20,30)(23,27)(24,25)(26,31)(29,32)(33,38,40,47)(34,42,44,45)(35,46,39,41)(36,48,43,37);(1,6,7,14)(2,9,5,13)(3,11,4,12)(8,15,10,16)(17,22)(18,21)(19,29)(20,24)(23,31)(25,30)(26,27)(28,32)(33,39,40,35)(34,43,44,36)(37,45,48,42)(38,41,47,46)Polycyclic presentation:
2^9|a^2=hi,b^2=h,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, 10319533);