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 | #50013 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3965 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)(3,10)(4,5)(6,7)(8,13)(9,14)(11,16)(12,15)(17,18,21,25)(19,23,27,22)(20,28,29,24)(26,30,32,31)(33,34,39,36)(35,42,41,38)(37,40,46,44)(43,48,47,45);(2,7)(4,14)(8,10)(11,15)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(33,34,39,36)(35,38,41,42)(37,40,46,44)(43,45,47,48);(17,18)(19,23)(20,24)(21,25)(22,27)(26,31)(28,29)(30,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,3)(2,8)(4,15)(5,16)(6,13)(7,10)(9,12)(11,14)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,36,39,34)(35,42,41,38)(37,40,46,44)(43,45,47,48);(1,4)(2,9)(3,11)(5,7)(6,14)(8,16)(10,12)(13,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,37,39,46)(34,40,36,44)(35,43,41,47)(38,45,42,48);(1,5,6,9)(2,4,7,14)(3,12,13,16)(8,11,10,15)(17,20,21,29)(18,24,25,28)(19,26,27,32)(22,30,23,31)(33,38,39,42)(34,41,36,35)(37,45,46,48)(40,47,44,43)Polycyclic presentation:
2^9|a^2=h,b^2=hi,d^2=hi,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10358851);