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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3928 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(5,13)(8,12)(10,16)(14,15)(17,18,23,27)(19,21,30,24)(20,26,22,25)(28,32,29,31)(33,34)(35,40)(36,45)(37,47)(38,39)(41,46)(42,43)(44,48);(2,4)(7,9)(8,12)(14,15)(17,19,23,30)(18,24,27,21)(20,28,22,29)(25,32,26,31)(34,39)(40,48)(41,45)(42,47);(1,2,6,4)(3,7,11,9)(5,8,13,12)(10,14,16,15)(17,20,23,22)(18,25,27,26)(19,28,30,29)(21,31,24,32)(33,35,38,44)(34,40,39,48)(36,43,46,37)(41,47,45,42);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,21,23,24)(18,19,27,30)(20,31,22,32)(25,28,26,29)(33,36)(34,41)(35,43)(37,44)(38,46)(39,45)(40,47)(42,48);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,19)(18,24)(20,28)(21,27)(22,29)(23,30)(25,32)(26,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,22,23,20)(18,26,27,25)(19,29,30,28)(21,32,24,31)(33,37,38,43)(34,42,39,47)(35,36,44,46)(40,41,48,45)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gi,d^2=hi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017817);