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 | |
3936 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)(19,26)(20,23)(21,28)(22,24)(25,29)(27,31)(30,32)(33,34,39,43)(35,37,46,40)(36,41,48,45)(38,44,47,42);(2,4)(7,9)(8,12)(14,15)(18,20)(21,29)(24,26)(27,30)(33,35,39,46)(34,40,43,37)(36,44,48,42)(38,45,47,41);(1,2,6,4)(3,7,11,9)(5,8,13,12)(10,14,16,15)(17,19,23,22)(18,24,20,26)(21,27,29,30)(25,31,28,32)(33,36,39,48)(34,41,43,45)(35,44,46,42)(37,47,40,38);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,20,23,18)(19,26,22,24)(21,28,29,25)(27,32,30,31)(33,37)(34,35)(36,47)(38,48)(39,40)(41,44)(42,45)(43,46);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,35)(34,37)(36,44)(38,45)(39,46)(40,43)(41,47)(42,48);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,22,23,19)(18,26,20,24)(21,30,29,27)(25,32,28,31)(33,38,39,47)(34,42,43,44)(35,45,46,41)(36,37,48,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,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, 8017818);