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 | C2×#120 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3941 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,8,12,16)(4,6,10,15)(5,14,13,9)(17,18,23,24)(19,25,28,30)(20,29,27,26)(21,31,32,22)(34,38)(35,41)(36,44)(46,48);(3,12)(5,13)(8,16)(9,14)(18,24)(21,32)(25,30)(26,29)(33,34,37,38)(35,42,41,39)(36,40,44,45)(43,48,47,46);(1,3,7,12)(2,8,11,16)(4,13,10,5)(6,9,15,14)(17,19,23,28)(18,25,24,30)(20,22,27,31)(21,26,32,29)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,4)(2,6)(3,13)(5,12)(7,10)(8,9)(11,15)(14,16)(17,20)(18,26)(19,22)(21,30)(23,27)(24,29)(25,32)(28,31)(33,35,37,41)(34,39,38,42)(36,43,44,47)(40,46,45,48);(1,5,7,13)(2,9,11,14)(3,4,12,10)(6,8,15,16)(17,21)(18,22)(19,26)(20,30)(23,32)(24,31)(25,27)(28,29)(33,36,37,44)(34,40,38,45)(35,43,41,47)(39,46,42,48);(1,6)(2,10)(3,9)(4,11)(5,8)(7,15)(12,14)(13,16)(17,22)(18,21)(19,27)(20,28)(23,31)(24,32)(25,26)(29,30)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=gi,d^2=h,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8400071);