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 | #2682 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3948 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,8,12,14)(4,9,13,16)(5,6,10,15)(17,18)(19,26)(20,23)(21,28)(22,24)(25,30)(27,31)(29,32)(33,34)(35,41)(36,43)(37,39)(38,42)(40,44)(45,48)(46,47);(1,3,7,12)(2,8,11,14)(4,6,13,15)(5,9,10,16)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,35,40,46)(34,41,44,47)(36,37,38,45)(39,42,48,43);(3,12)(4,13)(8,14)(9,16)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,36,40,38)(34,42,44,43)(35,45,46,37)(39,47,48,41);(1,4)(2,9)(3,6)(5,14)(7,13)(8,10)(11,16)(12,15)(17,20,23,18)(19,24,22,26)(21,28,30,25)(27,31,29,32)(33,37,40,45)(34,39,44,48)(35,36,46,38)(41,43,47,42);(1,5)(2,6)(3,9)(4,14)(7,10)(8,13)(11,15)(12,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,38,40,36)(34,43,44,42)(35,45,46,37)(39,47,48,41);(1,6)(2,10)(3,13)(4,12)(5,11)(7,15)(8,16)(9,14)(17,22,23,19)(18,24,20,26)(21,29,30,27)(25,31,28,32)(33,39)(34,37)(35,42)(36,47)(38,41)(40,48)(43,46)(44,45)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=h,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8619985);