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 |
| prime radical | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3708 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,13)(14,16)(33,34,38,39)(35,40,42,37)(36,44,46,41)(43,45,48,47);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,14)(13,16)(17,18,22,27)(19,25,28,23)(20,24,30,31)(21,26,32,29)(34,39)(35,42)(41,44)(43,48);(2,6)(3,5)(4,12)(15,16)(17,19,22,28)(18,23,27,25)(20,26,30,29)(21,31,32,24)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,45)(46,48);(1,4,7,12)(2,9,6,13)(3,11,5,14)(8,15,10,16)(17,20)(18,24)(19,26)(21,23)(22,30)(25,32)(27,31)(28,29)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,5,7,3)(2,10,6,8)(4,11,12,14)(9,15,13,16)(17,19)(18,25)(20,29)(21,24)(22,28)(23,27)(26,30)(31,32)(33,37,38,40)(34,35,39,42)(36,45,46,47)(41,43,44,48);(1,6,7,2)(3,10,5,8)(4,13,12,9)(11,16,14,15)(17,21)(18,26)(19,24)(20,23)(22,32)(25,30)(27,29)(28,31)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9405557);