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 | |
4115 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)(3,9)(4,8)(5,7)(6,13)(10,14)(11,15)(12,16)(17,18,23,26)(19,24,20,21)(22,25,30,32)(27,31,28,29)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,46)(47,48);(2,5)(8,9)(10,13)(15,16)(17,19,23,20)(18,21,26,24)(22,27,30,28)(25,29,32,31)(35,39)(36,43)(42,47)(44,48);(1,3,7,4)(2,8,5,9)(6,11,14,12)(10,15,13,16)(17,19,23,20)(18,24,26,21)(22,27,30,28)(25,31,32,29)(33,35,38,39)(34,36,41,43)(37,42,46,47)(40,44,45,48);(1,4,7,3)(2,9,5,8)(6,11,14,12)(10,15,13,16)(17,20,23,19)(18,24,26,21)(22,27,30,28)(25,29,32,31)(33,34)(35,36)(37,45)(38,41)(39,43)(40,46)(42,48)(44,47);(1,5,7,2)(3,9,4,8)(6,13,14,10)(11,16,12,15)(17,21,23,24)(18,20,26,19)(22,29,30,31)(25,28,32,27)(33,36)(34,39)(35,41)(37,44)(38,43)(40,47)(42,45)(46,48);(1,6)(2,10)(3,11)(4,12)(5,13)(7,14)(8,15)(9,16)(17,22,23,30)(18,25,26,32)(19,27,20,28)(21,29,24,31)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,45)(43,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10331812);