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 | |
4146 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)(19,23)(20,29)(21,31)(22,26)(24,30)(25,27)(28,32)(34,39)(35,43)(36,46)(37,47);(2,4)(7,9)(8,12)(14,15)(17,18,22,26)(19,23,28,32)(20,24,30,29)(21,25,27,31)(33,34,38,39)(35,40,43,48)(36,45,46,41)(37,42,47,44);(1,2,6,4)(3,7,11,9)(5,8,13,12)(10,14,16,15)(17,19,22,28)(18,23,26,32)(20,27,30,21)(24,31,29,25)(33,35)(34,40)(36,44)(37,45)(38,43)(39,48)(41,47)(42,46);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,20)(18,24)(19,27)(21,28)(22,30)(23,31)(25,32)(26,29)(33,36,38,46)(34,41,39,45)(35,44,43,42)(37,40,47,48);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,32)(29,30)(33,34)(35,40)(36,41)(37,42)(38,39)(43,48)(44,47)(45,46);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,21,22,27)(18,25,26,31)(19,20,28,30)(23,24,32,29)(33,37,38,47)(34,42,39,44)(35,45,43,41)(36,40,46,48)Polycyclic presentation:
2^9|b^2=g,c^2=ghi,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, 8017821);