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 | #46517 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(4,12)(7,9)(10,16)(17,18)(19,22)(20,27)(21,24)(23,28)(25,31)(26,30)(29,32)(33,34,39,45)(35,40,44,38)(36,43,47,41)(37,48,46,42);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31)(33,35,39,44)(34,40,45,38)(36,37,47,46)(41,42,43,48);(1,2,5,6)(3,7,11,9)(4,8,12,13)(10,14,16,15)(17,18,21,24)(19,22,26,30)(20,23,28,27)(25,29,32,31)(33,36,39,47)(34,41,45,43)(35,46,44,37)(38,42,40,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,37,39,46)(34,42,45,48)(35,36,44,47)(38,43,40,41);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32)(33,36)(34,43)(35,46)(37,44)(38,48)(39,47)(40,42)(41,45);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,20,21,28)(18,23,24,27)(19,25,26,32)(22,29,30,31)(33,38,39,40)(34,44,45,35)(36,42,47,48)(37,43,46,41)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=h,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9947740);