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 | |
4107 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)(3,11)(4,13)(5,10)(6,7)(8,14)(9,15)(12,16)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30);(2,7)(4,15)(8,11)(10,12)(17,19,22,27)(18,23,25,20)(21,26,29,32)(24,30,31,28)(33,34)(35,41)(36,44)(37,38)(39,43)(40,46)(42,47)(45,48);(2,7)(5,16)(8,11)(9,13)(21,29)(24,31)(26,32)(28,30)(34,38)(39,41)(40,44)(47,48);(1,3,6,14)(2,8,7,11)(4,10,15,12)(5,13,16,9)(17,20,22,23)(18,19,25,27)(21,28,29,30)(24,26,31,32)(33,35)(34,39)(36,45)(37,43)(38,41)(40,48)(42,46)(44,47);(1,4,6,15)(2,9,7,13)(3,12,14,10)(5,8,16,11)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,36)(34,40)(35,42)(37,46)(38,44)(39,47)(41,48)(43,45);(1,5,6,16)(2,10,7,12)(3,13,14,9)(4,8,15,11)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8292131);