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 | #57564 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3971 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,8,12,16)(4,5,9,15)(6,10,14,13)(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32)(33,34,38,41)(35,39,43,36)(37,40,46,48)(42,45,47,44);(1,3)(2,8)(4,13)(5,6)(7,12)(9,10)(11,16)(14,15)(18,24)(19,20)(26,29)(27,28)(33,35,38,43)(34,36,41,39)(37,42,46,47)(40,44,48,45);(3,12)(6,14)(8,16)(10,13)(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32)(33,36,38,39)(34,35,41,43)(37,45,46,44)(40,47,48,42);(1,4)(2,5)(3,13)(6,8)(7,9)(10,12)(11,15)(14,16)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,47)(46,48);(1,5)(2,9)(3,14)(4,11)(6,12)(7,15)(8,13)(10,16)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,45)(44,48)(46,47);(1,6,7,14)(2,10,11,13)(3,5,12,15)(4,8,9,16)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)(33,37,38,46)(34,40,41,48)(35,42,43,47)(36,44,39,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=g,d^2=h,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8281394);