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 | |
4110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,6,4)(5,9,15,16)(11,13,14,12)(17,18)(19,26)(20,23)(21,24)(22,25)(27,29)(28,31)(30,32)(33,34,37,41)(35,40,43,38)(36,44,46,39)(42,47,45,48);(1,3,7,6)(2,8,10,4)(5,11,15,14)(9,13,16,12)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,32)(30,31)(33,34)(35,38)(36,39)(37,41)(40,43)(42,47)(44,46)(45,48);(1,4,7,8)(2,3,10,6)(5,13,15,12)(9,14,16,11)(17,20,22,26)(18,23,25,19)(21,29,30,28)(24,27,32,31)(33,35,37,43)(34,38,41,40)(36,45,46,42)(39,48,44,47);(1,4)(2,3)(5,13)(6,10)(7,8)(9,14)(11,16)(12,15)(17,20)(18,19)(21,29)(22,26)(23,25)(24,31)(27,32)(28,30)(33,36,37,46)(34,39,41,44)(35,42,43,45)(38,47,40,48);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,13)(10,16)(17,21,22,30)(18,24,25,32)(19,27,23,31)(20,28,26,29)(33,35)(34,40)(36,45)(37,43)(38,41)(39,47)(42,46)(44,48);(1,6,7,3)(2,4,10,8)(5,14,15,11)(9,12,16,13)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(33,34)(35,40)(36,39)(37,41)(38,43)(42,48)(44,46)(45,47)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=gi,d^2=h,e^2=gh,f^2=i,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, 8936928);