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 | #39277 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3981 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,7,12,15)(4,5,13,8)(10,11,16,14)(17,18,22,26)(19,23,28,32)(20,29,30,24)(21,25,27,31)(34,38)(39,41)(40,46)(47,48);(1,3,6,12)(2,7,9,15)(4,10,13,16)(5,11,8,14)(17,19)(18,23)(20,21)(22,28)(24,31)(25,29)(26,32)(27,30)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,47)(45,46);(1,4,6,13)(2,8,9,5)(3,10,12,16)(7,14,15,11)(17,20,22,30)(18,24,26,29)(19,27,28,21)(23,25,32,31);(1,4)(2,5)(3,10)(6,13)(7,11)(8,9)(12,16)(14,15)(17,20)(18,24)(19,21)(22,30)(23,31)(25,32)(26,29)(27,28);(1,3)(2,7)(4,10)(5,11)(6,12)(8,14)(9,15)(13,16)(17,21)(18,25)(19,20)(22,27)(23,29)(24,32)(26,31)(28,30)(33,35,37,43)(34,39,38,41)(36,44,45,42)(40,48,46,47);(1,5)(2,4)(3,11)(6,8)(7,10)(9,13)(12,14)(15,16)(17,18)(19,23)(20,24)(21,31)(22,26)(25,27)(28,32)(29,30)(33,36,37,45)(34,40,38,46)(35,42,43,44)(39,47,41,48)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=h,e^2=g,f^2=g,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, 9820504);