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 | #42894 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4065 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,11,16)(4,6,14,15)(5,13,12,10)(17,18)(19,26)(20,23)(21,28)(22,25)(24,30)(27,29)(31,32)(36,44)(40,45)(42,48)(46,47);(1,3)(2,8)(4,5)(6,13)(7,11)(9,16)(10,15)(12,14)(18,20)(19,25)(24,28)(27,31)(34,38)(39,41)(40,45)(46,47);(2,9)(3,11)(4,14)(10,13)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,4)(2,6)(3,12)(5,11)(7,14)(8,10)(9,15)(13,16)(17,20,23,18)(19,26,25,22)(21,28,30,24)(27,32,31,29)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,5,7,12)(2,10,9,13)(3,4,11,14)(6,8,15,16)(17,21,23,30)(18,24,20,28)(19,27,25,31)(22,29,26,32);(1,6)(2,4)(3,13)(5,8)(7,15)(9,14)(10,11)(12,16)(17,22,23,26)(18,25,20,19)(21,29,30,32)(24,31,28,27)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=gh,d^2=ghi,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, 8014425);