Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of C32⋊H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C34 · Schur multiplier (H2) = C34 |
| direct product | C3×#6897 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C33 | C33 |
| commutator, Frattini, cosocle | C32 | C34 |
| prime radical | C34 | C32 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C33 ◁ G | |
| LECS | G ▷ C32 ▷ C1 | |
346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 243 |
A minimal faithful permutation representation — 4 generators on 57 points:
(1,2,3);(4,5,9)(6,13,26)(7,11,19)(8,12,20)(10,21,17)(14,23,29)(15,22,30)(16,24,27)(18,25,28)(31,32,35)(33,36,42)(34,38,43)(37,47,41)(39,44,50)(40,46,51)(45,54,49)(48,53,56)(52,57,55);(4,6,14,8,7,15,18,17,16)(5,10,22,12,13,24,25,11,23)(9,19,27,20,21,29,28,26,30)(31,33,39,34,40,48,41,49,55)(32,36,44,38,46,53,37,45,52)(35,42,50,43,51,56,47,54,57);(4,7,16,8,17,14,18,6,15)(5,11,24,12,10,23,25,13,22)(9,19,27,20,21,29,28,26,30)(32,37,38)(35,43,47)(36,45,46)(42,51,54)(44,52,53)(50,56,57)153 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 434);