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 | |
4288 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)(3,10)(4,7)(5,13)(6,8)(9,14)(11,15)(12,16)(17,18)(19,22)(20,29)(21,25)(23,26)(24,28)(27,32)(30,31)(33,34)(35,39)(36,37)(38,42)(40,45)(41,48)(43,44)(46,47);(1,3,6,4)(2,7,8,10)(5,11,14,12)(9,15,13,16)(17,19)(18,22)(20,21)(23,27)(24,30)(25,29)(26,32)(28,31)(33,35)(34,39)(36,43)(37,44)(38,45)(40,42)(41,46)(47,48);(2,8)(5,14)(7,10)(11,12)(19,27)(20,28)(22,32)(24,29)(33,35,38,45)(34,40,42,39)(36,43,41,46)(37,47,48,44);(1,2)(3,7)(4,10)(5,9)(6,8)(11,15)(12,16)(13,14)(17,20)(18,24)(19,21)(22,30)(23,28)(25,32)(26,29)(27,31)(33,35)(34,40)(36,46)(37,44)(38,45)(39,42)(41,43)(47,48);(1,4,6,3)(2,7,8,10)(5,12,14,11)(9,15,13,16)(17,21)(18,25)(19,28)(20,27)(22,24)(23,31)(26,30)(29,32)(33,36,38,41)(34,37,42,48)(35,43,45,46)(39,44,40,47);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,13)(10,16)(17,22)(18,19)(20,30)(21,24)(23,32)(25,28)(26,27)(29,31)(33,37)(34,41)(35,44)(36,42)(38,48)(39,46)(40,43)(45,47)Polycyclic presentation:
2^9|b^2=g,c^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8834897);