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 |
| prime radical | #46232 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,4,11,7)(5,13,14,8)(10,15,16,12)(18,23)(21,31)(24,30)(25,32)(36,43)(40,45)(42,47)(46,48);(1,3,6,11)(2,7,9,4)(5,10,14,16)(8,15,13,12)(19,27)(20,29)(24,30)(25,32)(34,38)(35,41)(40,45)(42,47);(17,18,22,23)(19,24,27,30)(20,25,29,32)(21,26,31,28)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,4)(2,3)(5,12)(6,7)(8,10)(9,11)(13,16)(14,15)(17,19)(18,24)(20,28)(21,25)(22,27)(23,30)(26,29)(31,32);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,15)(14,16)(17,20,22,29)(18,25,23,32)(19,28,27,26)(21,30,31,24)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,5,6,14)(2,8,9,13)(3,10,11,16)(4,12,7,15)(17,21)(18,26)(19,25)(20,30)(22,31)(23,28)(24,29)(27,32)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000424);