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 | |
4229 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)(3,5)(4,7)(6,12)(8,10)(9,14)(11,15)(13,16)(17,18,23,24)(19,25,20,21)(22,26,31,32)(27,30,28,29)(33,34,38,41)(35,39,44,47)(36,37,45,40)(42,43,48,46);(1,3)(2,8)(4,5)(6,11)(7,10)(9,15)(12,13)(14,16)(18,24)(21,25)(26,32)(29,30)(33,35,38,44)(34,39,41,47)(36,42,45,48)(37,43,40,46);(2,4)(3,10)(9,12)(11,16)(17,19,23,20)(18,25,24,21)(22,27,31,28)(26,30,32,29)(33,36,38,45)(34,40,41,37)(35,42,44,48)(39,46,47,43);(1,4,7,2)(3,5,10,8)(6,12,14,9)(11,13,16,15)(17,20,23,19)(18,21,24,25)(22,28,31,27)(26,29,32,30)(33,36)(34,37)(35,42)(38,45)(39,43)(40,41)(44,48)(46,47);(1,5)(2,3)(4,10)(6,13)(7,8)(9,11)(12,16)(14,15)(17,21)(18,19)(20,24)(22,30)(23,25)(26,28)(27,32)(29,31)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(1,6,7,14)(2,9,4,12)(3,11,10,16)(5,13,8,15)(17,22,23,31)(18,26,24,32)(19,27,20,28)(21,29,25,30)(33,37)(34,36)(35,43)(38,40)(39,42)(41,45)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9820501);