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 | #2714 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3892 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)(3,9)(4,13)(5,7)(6,10)(8,11)(12,14)(15,16)(17,18,23,21)(19,26,28,24)(20,29,22,25)(27,31,30,32)(33,34)(35,38)(36,42)(37,47)(39,41)(40,43)(44,48)(45,46);(1,3,8,5)(2,9,11,7)(4,12,6,15)(10,16,13,14)(17,19)(18,24)(20,27)(21,26)(22,30)(23,28)(25,31)(29,32)(33,35,40,37)(34,38,43,47)(36,45,39,44)(41,48,42,46);(1,4,8,6)(2,10,11,13)(3,12,5,15)(7,14,9,16)(17,20,23,22)(18,25,21,29)(19,27,28,30)(24,31,26,32)(33,36,40,39)(34,41,43,42)(35,44,37,45)(38,46,47,48);(1,5,8,3)(2,7,11,9)(4,12,6,15)(10,16,13,14)(17,21,23,18)(19,26,28,24)(20,25,22,29)(27,31,30,32)(33,37,40,35)(34,38,43,47)(36,44,39,45)(41,48,42,46);(1,6,8,4)(2,10,11,13)(3,12,5,15)(7,16,9,14)(17,22,23,20)(18,25,21,29)(19,27,28,30)(24,32,26,31)(33,38)(34,35)(36,46)(37,43)(39,48)(40,47)(41,44)(42,45);(1,7)(2,3)(4,14)(5,11)(6,16)(8,9)(10,12)(13,15)(17,19)(18,24)(20,27)(21,26)(22,30)(23,28)(25,31)(29,32)(33,39,40,36)(34,42,43,41)(35,44,37,45)(38,46,47,48)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9620067);