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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3890 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)(3,11)(4,7)(5,9)(6,10)(8,13)(12,14)(15,16)(17,18,23,26)(19,27,29,20)(21,24,30,28)(22,25,31,32);(1,3,7,13)(2,8,4,11)(5,6,12,16)(9,15,14,10)(17,19,23,29)(18,20,26,27)(21,25,30,32)(22,28,31,24)(34,38)(39,45)(40,46)(47,48);(1,2,7,4)(3,11,13,8)(5,9,12,14)(6,10,16,15)(17,20,23,27)(18,19,26,29)(21,22,30,31)(24,25,28,32)(33,34)(35,39)(36,40)(37,38)(41,47)(42,45)(43,48)(44,46);(1,4,7,2)(3,8,13,11)(5,14,12,9)(6,15,16,10)(17,18)(19,20)(21,24)(22,25)(23,26)(27,29)(28,30)(31,32);(1,5)(2,9)(3,6)(4,14)(7,12)(8,15)(10,11)(13,16)(17,21)(18,24)(19,25)(20,22)(23,30)(26,28)(27,31)(29,32)(33,35,37,42)(34,39,38,45)(36,43,44,41)(40,48,46,47);(1,6)(2,10)(3,12)(4,15)(5,13)(7,16)(8,9)(11,14)(17,22)(18,25)(19,28)(20,30)(21,27)(23,31)(24,29)(26,32)(33,36,37,44)(34,40,38,46)(35,41,42,43)(39,47,45,48)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=h,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8003765);