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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,5)(3,8,6,13)(4,9,11,14)(10,15,12,16)(17,18)(19,26)(20,28)(21,25)(22,23)(24,29)(27,31)(30,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,45)(44,47)(46,48);(1,3,7,6)(2,8,5,13)(4,10,11,12)(9,15,14,16)(17,19,23,25)(18,21,22,26)(20,27,29,32)(24,31,28,30)(33,35)(34,36)(37,43)(38,39)(40,44)(41,42)(45,48)(46,47);(2,5)(3,6)(4,11)(15,16)(18,22)(20,29)(21,26)(27,32)(33,36,38,42)(34,39,41,35)(37,45,46,44)(40,43,48,47);(1,4)(2,9)(3,10)(5,14)(6,12)(7,11)(8,15)(13,16)(17,20,23,29)(18,24,22,28)(19,27,25,32)(21,31,26,30)(33,36)(34,35)(37,45)(38,42)(39,41)(40,47)(43,48)(44,46);(1,5,7,2)(3,8,6,13)(4,9,11,14)(10,16,12,15)(17,21,23,26)(18,25,22,19)(20,30,29,31)(24,27,28,32)(33,37,38,46)(34,40,41,48)(35,43,39,47)(36,44,42,45);(1,6,7,3)(2,8,5,13)(4,12,11,10)(9,15,14,16)(17,22,23,18)(19,26,25,21)(20,28,29,24)(27,30,32,31)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=ghi,d^2=hi,e^2=gi,f^2=h,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, 8922740);