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 | #52962 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4087 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)(3,7,8,4)(5,13,14,9)(11,12,16,15)(17,18)(19,21)(20,28)(22,23)(24,26)(25,30)(27,31)(29,32)(36,44)(40,45)(42,48)(46,47);(1,3)(2,7)(4,10)(5,11)(6,8)(9,15)(12,13)(14,16)(18,23)(19,26)(25,28)(27,32)(34,38)(39,41)(40,45)(46,47);(1,4,6,7)(2,8,10,3)(5,12,14,15)(9,16,13,11)(17,18,22,23)(19,24,26,21)(20,25,30,28)(27,31,32,29)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,4)(2,3)(5,12)(6,7)(8,10)(9,11)(13,16)(14,15)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,5,6,14)(2,9,10,13)(3,11,8,16)(4,12,7,15)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,15)(14,16)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=gh,c^2=h,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014471);