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 | #52768 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3911 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,5)(4,12)(7,10)(9,15)(20,28)(24,29)(26,32)(30,31)(33,34)(35,41)(36,38)(37,40)(39,43)(42,47)(44,45)(46,48);(1,2,6,5)(3,7,11,10)(4,8,12,13)(9,14,15,16)(18,22)(23,25)(24,29)(30,31)(34,36)(39,41)(40,44)(46,47);(1,3,6,11)(2,7,5,10)(4,9,12,15)(8,14,13,16)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,4)(2,8)(3,9)(5,13)(6,12)(7,14)(10,16)(11,15)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(1,5,6,2)(3,10,11,7)(4,13,12,8)(9,16,15,14)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|b^2=gh,c^2=gi,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8002223);