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 | |
4080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,8,13,7)(4,5,9,15)(11,16,14,12)(17,18)(19,22)(20,29)(21,25)(23,26)(24,28)(27,32)(30,31)(33,34)(35,40)(36,39)(37,41)(38,48)(42,45)(43,44)(46,47);(1,3,6,13)(2,7,10,8)(4,11,9,14)(5,12,15,16)(17,19)(18,22)(20,21)(23,27)(24,30)(25,29)(26,32)(28,31)(33,35)(34,40)(36,44)(37,45)(38,46)(39,43)(41,42)(47,48);(4,9)(5,15)(11,14)(12,16)(19,27)(20,28)(22,32)(24,29)(34,36)(35,43)(38,45)(41,47);(1,4,6,9)(2,5,10,15)(3,11,13,14)(7,12,8,16)(17,20)(18,24)(19,21)(22,30)(23,28)(25,32)(26,29)(27,31)(33,36,39,34)(35,44,43,40)(37,41,46,47)(38,48,45,42);(1,3)(2,8)(4,14)(5,12)(6,13)(7,10)(9,11)(15,16)(17,21)(18,25)(19,28)(20,27)(22,24)(23,31)(26,30)(29,32)(33,37,39,46)(34,41,36,47)(35,45,43,38)(40,42,44,48);(1,5)(2,9)(3,12)(4,10)(6,15)(7,14)(8,11)(13,16)(17,22)(18,19)(20,30)(21,24)(23,32)(25,28)(26,27)(29,31)(33,38,39,45)(34,42,36,48)(35,46,43,37)(40,41,44,47)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=gh,e^2=h,f^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, 8835793);