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 | #2688 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4019 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)(3,8,11,15)(4,6,13,16)(5,10,12,14)(17,18,23,27)(19,24,28,31)(20,25,30,29)(21,22,26,32)(33,34,39,44)(35,45,48,40)(36,41,47,42)(37,43,46,38);(1,3,7,11)(2,8,9,15)(4,12,13,5)(6,14,16,10)(17,19)(18,24)(20,22)(21,29)(23,28)(25,26)(27,31)(30,32)(33,35)(34,40)(36,46)(37,47)(38,42)(39,48)(41,43)(44,45);(2,9)(3,11)(4,13)(10,14)(19,28)(20,30)(24,31)(25,29)(33,36,39,47)(34,41,44,42)(35,46,48,37)(38,40,43,45);(1,4)(2,6)(3,12)(5,11)(7,13)(8,14)(9,16)(10,15)(17,20)(18,25)(19,22)(21,31)(23,30)(24,26)(27,29)(28,32)(33,36)(34,42)(35,46)(37,48)(38,40)(39,47)(41,44)(43,45);(1,5,7,12)(2,10,9,14)(3,13,11,4)(6,8,16,15)(17,21)(18,22)(19,29)(20,24)(23,26)(25,28)(27,32)(30,31)(33,37,39,46)(34,43,44,38)(35,47,48,36)(40,41,45,42);(1,6)(2,4)(3,14)(5,15)(7,16)(8,12)(9,13)(10,11)(17,22)(18,26)(19,20)(21,27)(23,32)(24,25)(28,30)(29,31)(33,38,39,43)(34,37,44,46)(35,42,48,41)(36,40,47,45)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9034727);