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 | #54933 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,10,7)(4,12,13,8)(11,14,16,15)(17,18,23,26)(19,25,22,24)(20,29,30,21)(27,28,31,32)(33,34)(35,40)(36,38)(37,44)(39,42)(41,45)(43,46)(47,48);(2,6)(7,9)(8,12)(14,15)(17,19,23,22)(18,24,26,25)(20,27,30,31)(21,28,29,32)(35,42)(39,40)(43,48)(46,47);(2,6)(3,10)(8,12)(11,16)(20,30)(21,29)(27,31)(28,32)(33,35,38,42)(34,39,36,40)(37,43,45,48)(41,46,44,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20)(18,21)(19,27)(22,31)(23,30)(24,28)(25,32)(26,29)(33,35)(34,40)(36,39)(37,43)(38,42)(41,47)(44,46)(45,48);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,21)(18,20)(19,28)(22,32)(23,29)(24,27)(25,31)(26,30)(33,36,38,34)(35,39,42,40)(37,44,45,41)(43,46,48,47);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,22,23,19)(18,25,26,24)(20,31,30,27)(21,32,29,28)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034605);