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 | #49797 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4127 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,8)(9,14)(10,15)(33,34,38,39)(35,43,46,40)(36,41,47,45)(37,44,48,42);(1,2,6,7)(3,4,8,11)(5,9,13,14)(10,12,15,16)(17,18,22,21)(19,23,25,27)(20,24,28,30)(26,31,29,32)(34,39)(36,47)(40,43)(42,44);(3,8)(4,11)(5,13)(9,14)(18,21)(19,25)(20,28)(31,32)(33,35,38,46)(34,40,39,43)(36,42,47,44)(37,41,48,45);(1,3)(2,4)(5,10)(6,8)(7,11)(9,12)(13,15)(14,16)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,28)(30,31)(33,35)(34,40)(36,42)(37,45)(38,46)(39,43)(41,48)(44,47);(1,4)(2,8)(3,7)(5,12)(6,11)(9,15)(10,14)(13,16)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,29)(27,32)(33,36)(34,41)(35,44)(37,40)(38,47)(39,45)(42,46)(43,48);(1,5)(2,9)(3,10)(4,12)(6,13)(7,14)(8,15)(11,16)(17,21,22,18)(19,27,25,23)(20,30,28,24)(26,32,29,31)(33,37)(34,42)(35,45)(36,40)(38,48)(39,44)(41,46)(43,47)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8929134);