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 | |
3969 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,9,10,6)(4,7,13,12)(11,15,16,14)(17,18,23,25)(19,21,27,20)(22,24,31,30)(26,29,32,28)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,46)(47,48);(3,10)(6,9)(11,16)(14,15)(17,19,23,27)(18,20,25,21)(22,26,31,32)(24,28,30,29)(35,39)(36,43)(42,47)(44,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,20,23,21)(18,19,25,27)(22,28,31,29)(24,26,30,32)(33,35,38,39)(34,36,41,43)(37,42,46,47)(40,44,45,48);(1,4)(2,7)(3,11)(5,13)(6,14)(8,12)(9,15)(10,16)(17,18)(19,20)(21,27)(22,30)(23,25)(24,31)(26,29)(28,32)(33,34)(35,36)(37,45)(38,41)(39,43)(40,46)(42,48)(44,47);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,21,23,20)(18,19,25,27)(22,29,31,28)(24,26,30,32)(33,36)(34,39)(35,41)(37,44)(38,43)(40,47)(42,45)(46,48);(1,2)(3,6)(4,12)(5,8)(7,13)(9,10)(11,15)(14,16)(17,22)(18,24)(19,26)(20,28)(21,29)(23,31)(25,30)(27,32)(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=g,c^2=ghi,e^2=g,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, 10330332);