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 | |
4311 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)(3,10)(4,13)(5,8)(6,14)(7,9)(11,12)(15,16)(18,21)(19,20)(22,30)(31,32)(35,42)(37,45)(39,40)(41,44);(1,3,7,5)(2,8,9,10)(4,11,14,15)(6,12,13,16)(17,18,23,21)(19,24,20,26)(22,25,30,29)(27,31,28,32)(33,34)(35,40)(36,38)(37,41)(39,42)(43,47)(44,45)(46,48);(17,19,23,20)(18,24,21,26)(22,27,30,28)(25,31,29,32)(33,35)(34,39)(36,40)(37,43)(38,42)(41,46)(44,47)(45,48);(1,4)(2,6)(3,11)(5,15)(7,14)(8,12)(9,13)(10,16)(17,20,23,19)(18,24,21,26)(22,28,30,27)(25,31,29,32)(33,36,38,34)(35,40,42,39)(37,44,45,41)(43,47,48,46);(1,5,7,3)(2,8,9,10)(4,11,14,15)(6,16,13,12)(17,21,23,18)(19,26,20,24)(22,29,30,25)(27,32,28,31)(33,35)(34,40)(36,39)(37,43)(38,42)(41,47)(44,46)(45,48);(1,6)(2,4)(3,12)(5,16)(7,13)(8,11)(9,14)(10,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|b^2=g,c^2=hi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8044803);