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 | #50808 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4129 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,11,15)(4,13,14,8)(5,12,16,9)(17,18,22,26)(19,23,28,27)(20,30,31,24)(21,25,32,29)(33,34)(35,41)(36,44)(37,40)(38,43)(39,45)(42,46)(47,48);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,15)(12,14)(17,18)(19,27)(20,30)(21,25)(22,26)(23,28)(24,31)(29,32)(33,34,37,40)(35,38,43,41)(36,39,45,44)(42,46,48,47);(3,11)(4,14)(7,15)(8,13)(19,28)(21,32)(23,27)(25,29)(33,34,37,40)(35,38,43,41)(36,44,45,39)(42,47,48,46);(1,4)(2,8)(3,12)(5,7)(6,14)(9,11)(10,13)(15,16)(17,19,22,28)(18,23,26,27)(20,25,31,29)(21,30,32,24)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,5)(2,9)(3,8)(4,15)(6,16)(7,14)(10,12)(11,13)(17,20,22,31)(18,24,26,30)(19,29,28,25)(21,23,32,27)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,3)(2,7)(4,12)(5,8)(6,11)(9,14)(10,15)(13,16)(17,21,22,32)(18,25,26,29)(19,30,28,24)(20,23,31,27)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=h,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8841524);