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 | #2671 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3814 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)(3,6)(4,11)(5,8)(7,12)(9,13)(10,16)(14,15)(17,18)(19,24)(20,23)(21,25)(22,26)(27,32)(28,31)(29,30)(33,34,37,40)(35,38,41,45)(36,39,44,48)(42,46,43,47);(3,9)(6,13)(10,15)(14,16)(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32)(35,41)(38,45)(42,43)(46,47);(1,2,5,8)(3,6,9,13)(4,7,12,11)(10,14,15,16)(17,19,22,20)(18,24,26,23)(21,27,30,28)(25,32,29,31)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,44)(47,48);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,20,22,19)(18,23,26,24)(21,28,30,27)(25,31,29,32)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,47)(44,48);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,18)(19,24)(20,23)(21,29)(22,26)(25,30)(27,31)(28,32)(33,35)(34,38)(36,43)(37,41)(39,47)(40,45)(42,44)(46,48);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,36)(34,39)(35,42)(37,44)(38,46)(40,48)(41,43)(45,47)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gh,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9820126);