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 | #2410 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4066 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)(3,8,12,14)(4,9,13,16)(5,6,10,15)(17,18,23,28)(19,29,31,24)(20,22,27,25)(21,32,30,26)(33,34)(35,41)(36,47)(37,38)(39,43)(40,44)(42,46)(45,48);(1,3,7,12)(2,8,11,14)(4,6,13,15)(5,9,10,16)(17,19)(18,24)(20,21)(22,26)(23,31)(25,32)(27,30)(28,29)(33,35,40,47)(34,41,44,36)(37,43,42,45)(38,39,46,48);(3,12)(4,13)(8,14)(9,16)(17,20,23,27)(18,25,28,22)(19,21,31,30)(24,32,29,26)(33,36,40,41)(34,35,44,47)(37,48,42,39)(38,43,46,45);(1,4)(2,9)(3,6)(5,14)(7,13)(8,10)(11,16)(12,15)(17,20)(18,25)(19,30)(21,31)(22,28)(23,27)(24,26)(29,32)(33,37,40,42)(34,38,44,46)(35,45,47,43)(36,39,41,48);(1,5)(2,6)(3,9)(4,14)(7,10)(8,13)(11,15)(12,16)(17,21,23,30)(18,26,28,32)(19,20,31,27)(22,29,25,24)(33,38)(34,42)(35,39)(36,43)(37,44)(40,46)(41,45)(47,48);(1,6)(2,10)(3,13)(4,12)(5,11)(7,15)(8,16)(9,14)(17,22,23,25)(18,27,28,20)(19,26,31,32)(21,24,30,29)(33,39)(34,43)(35,46)(36,37)(38,47)(40,48)(41,42)(44,45)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=h,d^2=gh,e^2=g,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, 8619529);