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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4042 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)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18)(19,25)(20,21)(22,24)(23,29)(26,31)(27,30)(28,32)(33,34)(35,38)(36,39)(37,40)(41,42)(43,48)(44,47)(45,46);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,22,29)(18,23,24,25)(20,27,26,32)(21,28,31,30)(33,35)(34,38)(36,42)(37,45)(39,41)(40,46)(43,47)(44,48);(2,3)(7,9)(8,10)(14,15)(18,24)(20,26)(23,25)(27,32)(34,36)(35,41)(40,44)(43,45);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,22,26)(18,21,24,31)(19,27,29,32)(23,28,25,30)(33,36,39,34)(35,42,41,38)(37,44,47,40)(43,46,45,48);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,19)(18,25)(20,27)(21,30)(22,29)(23,24)(26,32)(28,31)(33,37,39,47)(34,40,36,44)(35,43,41,45)(38,48,42,46);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,21)(18,26)(19,28)(20,24)(22,31)(23,32)(25,27)(29,30)(33,38,39,42)(34,41,36,35)(37,46,47,48)(40,43,44,45)Polycyclic presentation:
2^9|b^2=g,d^2=h,e^2=gh,f^2=gh,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, 9619704);