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 | |
4312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,6,4,8)(5,10,14,16)(11,13,12,15)(17,18)(19,25)(20,28)(21,24)(22,26)(23,30)(27,31)(29,32)(33,34,38,39)(35,41,36,40)(37,45,46,42)(43,47,44,48);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(17,18,21,24)(19,22,26,25)(20,23,30,28)(27,31,29,32)(34,39)(37,46)(40,41)(43,44);(2,9)(3,4)(10,16)(11,12)(19,26)(22,25)(27,29)(31,32)(33,35,38,36)(34,40,39,41)(37,43,46,44)(42,47,45,48);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(17,19,21,26)(18,22,24,25)(20,29,30,27)(23,32,28,31)(33,36,38,35)(34,41,39,40)(37,43,46,44)(42,47,45,48);(1,5)(2,10)(3,11)(4,12)(6,13)(7,14)(8,15)(9,16)(17,20)(18,23)(19,27)(21,30)(22,31)(24,28)(25,32)(26,29)(33,37)(34,42)(35,43)(36,44)(38,46)(39,45)(40,47)(41,48);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(17,18)(19,22)(20,23)(21,24)(25,26)(27,31)(28,30)(29,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=hi,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9297746);