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 | |
4119 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,12,8)(4,9,10,5)(13,14,16,15)(17,18)(19,26)(20,23)(21,24)(22,28)(25,30)(27,31)(29,32)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(2,7)(3,12)(5,9)(13,16)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,34)(35,41)(36,44)(37,40)(38,43)(39,45)(42,46)(47,48);(2,7)(4,10)(8,11)(13,16)(17,18,23,20)(19,26,21,24)(22,25,30,28)(27,32,29,31)(33,34,37,40)(35,38,43,41)(36,39,45,44)(42,46,48,47);(1,3)(2,8)(4,13)(5,14)(6,12)(7,11)(9,15)(10,16)(17,20,23,18)(19,26,21,24)(22,28,30,25)(27,32,29,31)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,4,6,10)(2,9,7,5)(3,13,12,16)(8,15,11,14)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,5,6,9)(2,10,7,4)(3,14,12,15)(8,16,11,13)(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=i,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10275855);